A h-principle for open relations invariant under foliated isotopies

Mélanie Bertelson ^{∗}

November 17, 2000

Abstract This paper presents a natural extension to foliated spaces of the following result due to Gromov : the h-principle for open, invariant differential relations is valid on open manifolds. The definition of openness for foliated spaces adopted here involves a certain type of Morse functions. Consequences concerning the problem of existence of regular Poisson structures, the original motivation for this work, are presented.

Introduction

Gromov proved in [9] the following theorem.

Theorem 0.1 On an open manifold, the h-principle for open, invariant differential relations is valid.

Heuristically, a differential relation on a manifold M is a differential constraint on the sections of a certain bundle π : E → M. More precisely, it is a subset Ω of some jet bundle J^{k}(E) of local sections of E. A section of E whose kjet extension is entirely contained in the relation satisfies the constraint; it is called a solution of the differential relation Ω. A relation Ω is said to be open (respectively invariant) when Ω is an open subset of J^{k}(E) (respectively when isotopies of M can be lifted to isotopies of J^{k}(E) that preserve Ω). Letting Sol(Ω) (respectively Γ(Ω)) denote the set of solutions (respectively sections) of Ω, endowed with the weak C^{k} (respectively C^{0}) topology, the relation Ω is said to satisfy the h-principle if the k-jet extension map

j^{k} : Sol(Ω) → Γ(Ω) : f ↦ j^{k}f

is a weak homotopy equivalence. This means that the map j^{k} induces a bijection inbetween arcwise connected components of Sol(Ω) and Γ(Ω), and isomorphisms of the homotopy groups for the various components.

∗Max-Planck-Institut für Mathematik, D-53111 Bonn, Germany (bertel@mpimbonn.mpg.de). This work has been supported by an Alfred P. Sloan Dissertation Fellowship.

Theorem 0.1 has a lot of important corollaries concerning the existence and classification problems of various types of objects in differential topology and geometry (cf. [10]). For instance, applying Theorem 0.1 to the relation Ω^{S} = { j^{1}α(x) ∈ J^{1}(T^{∗}M); dα(x) is nondegenerate } yields the following result.

Corollary 0.2 Let M be an open manifold. The inclusion of the space of exact symplectic forms on M into the space of nondegenerate 2-forms is a weak homotopy equivalence.

In particular, on an open manifold, existence of a symplectic structure depends only on existence of a nondegenerate 2-form, a problem that belongs to obstruction theory.

The hypothesis that M is open is crucial. Indeed, already for the symplectic relation Ω^{S}, the h-principle is far from being valid on a closed manifold. In addition to a nondegenerate 2-form, a closed symplectic manifold admits a de Rham class in H^{2}(M) whose top exterior power does not vanish. More subtle conditions, involving Seiberg–Witten invariants, have been discovered by Taubes (cf. [19]). Furthermore, even when the manifold admits a symplectic structure, not any nondegenerate 2-form may be deformed into a symplectic form. In general, if every open, invariant differential relation defined on a manifold M satisfies the h-principle, then M must be open, as such a relation can be constructed that admits sections and whose solutions are functions without local maxima.

Motivated by the problem of existence of leafwise symplectic structures on foliated spaces (cf. [1, 2]), we searched for a generalization of Theorem 0.1 to foliated invariant differential relations, that is, differential relations that are invariant under isotopies that preserve a certain foliation on the manifold. This requires finding a good notion of “openness” for foliated spaces. It is important to observe that one may not, in general, impose on the solutions constructed in the proof of Theorem 0.1 to be nicely behaved at infinity. In contrast, a solution of a differential relation (a symplectic structure for instance) on a nonclosed leaf of a foliation, that is the restriction of a global solution (a leafwise symplectic structure) is most likely very constrained at infinity, partly due to recurrence phenomena, partly due to the influence of neighboring leaves. The foliated case lies, in some sense, midway between the open case and the closed case. Examples of foliations have been exhibited in [1, 2] that seem from some point of view quite open, that support leafwise nondegenerate 2-forms, but do not admit any leafwise symplectic structure. One should therefore avoid to include them in the class of open foliations.

On the other hand, open manifolds are characterized by the existence of a positive, proper Morse function, without any local maximum. The proof of Theorem 0.1 suggests to base a definition of openness for foliated manifolds on that characterization. This justifies the following definition.

Definition 0.3 A foliated manifold (M, F) is said to be open if there exists a smooth function f : M → [0, ∞) that has the following properties :

a) f is proper,

b) f has no leafwise local maxima,

c) f is F-generic (cf. Definition 3.21 below).

With this definition of openness, the following result holds :

Theorem 0.4 On an open foliated manifold, any open, foliated invariant differential relation satisfies the parametric h-principle.

Corollary 0.5 Let (M, F) be an open foliated manifold. Any leafwise nondegenerate 2-form is homotopic, in the class of leafwise nondegenerate 2-forms, to a leafwise symplectic form.

The leaves of an open foliation are necessarily open manifolds, but this condition is not sufficient. One might interpret Definition 0.3 as that of uniform openness of the leaves. It can be checked directly (i.e. without quoting Corollary 0.5) that the foliated manifolds introduced in [1, 2] do not support any function f satisfying a) and b).

The proof of Theorem 0.4 involves consideration of the trajectories of a leafwise gradient vector field for f, as did the proof of Theorem 0.1. There are new difficulties. First, the leafwise critical points of f are not isolated but come in families. Thus they cannot be handled one at a time (as they are in the nonfoliated case). Secondly, leafwise critical points may be degenerate, even generically. The set of trajectories converging to a degenerate critical point is not in general well understood. To overcome the latter difficulty one needs to carry out a careful construction of a Riemannian metric for which the trajectories of the associated leafwise gradient vector field are somewhat controlled near the singular locus.

This paper is organized as follows. Section 1 is a brief introduction to the theory of h-principles (see [10]). Section 2 presents an outline of the proof of Theorem 0.4, and should serve as a reading-guide for the remainder of the text. Section 3 is devoted to making precise the term F-generic referred to in Definition 0.3, and to introducing the notion of strong F-genericity needed later. Section 4 exhibits some properties of leafwise gradient vector fields (Definition 4.1) of strongly F-generic functions. Section 5 describes the construction of a nice Riemannian metric associated to a strongly F-generic function. The proof of Theorem 0.4 is completed in Section 6. Some examples of open foliated manifolds are presented in Section 7. Finally, Section 8 is concerned with Corollary 0.5.

Acknowledgements. I wish to thank Emmanuel Giroux, who suggested to generalize Gromov’s result, Alan Weinstein, whose insightful advice has helped this work reach its final shape, and Yasha Eliashberg for many very stimulating conversations.

1 H-principles

We state here the definitions and results of the theory of h-principles (cf. [10]) that will be needed in the remainder of the text. We have followed Emmanuel Giroux’s beautiful (as yet unpublished) lecture notes [7]. The proofs are reproduced in [1].

1.1 Differential relations and h-principles

Consider a locally trivial fibration π : E → M with fiber a manifold F. Let E^{k} denote the set of k-jets of local sections of π : E → M. If f : U ⊂ M → E is a local section defined on an open subset U of M, the k-jet of f at x ∈ U is denoted by j^{k}f(x). The set E^{k}, endowed with the natural projection E^{k} → M : j^{k}f(x) ↦ x, is a locally trivial fibration. For k ≤ r ≤ ∞, a C^{r} local section f of E, defined on an open subset U, induces a C^{r−k} local section j^{k}f : U → E^{k} : x ↦ j^{k}f(x) of E^{k}, called the k-jet extension of f.

Definition 1.1 A subset Ω of E^{k} is called a differential relation of order k. It is said to be open if it is an open subset. A section of E^{k} whose values are in Ω is called a section of Ω. A C^{k} section of E whose k-jet extension is a section of Ω is called a solution of Ω. A section of E^{k} that coincides with the k-jet extension of some section of E is said to be holonomic.

Observe that solutions of Ω and holonomic sections of Ω are in one to one correspondence.

Let Ω be a differential relation on the manifold M. We will be considering families of sections of Ω parameterized by cubes S = [0, 1]^{p},p ≥ 1. Those are defined to be continuous maps f : M × S → E^{k}, such that for each s in S, the partial map f_{s} : M → E^{k} (obtained by restricting the map f to M × {s}) is a section of Ω. A homotopy of sections of Ω is a family parameterized by S = [0, 1]. Concerning local sections of Ω, we need to introduce some terminology. Let (A, A′) denote a nested pair of compact subsets of M. The word “nested” indicates that A′ ⊂ A.

- A family of sections of Ω defined near A is a family defined on U × S for some neighborhood U of A.

- Two families of sections f_{s} and g_{s} are said to coincide near A if there exists a neighborhood U of A on which both f_{s} and g_{s} are defined, and for which f_{s}|_{U} = g_{s}|_{U} for all s in S.

- A family of sections g_{s} defined near A is said to extend another family f_{s} defined near A′ if g_{s} and f_{s} coincide near A′.

- Two families of sections f_{s} and g_{s} defined near A are said to be homotopic if there exists a third family h_{s,t} defined near A with (s, t) in S × [0, 1], such that h_{s,0} coincides with f_{s} near A and h_{s,1} coincides with g_{s} near A.

- A homotopy h_{s,t} is said to be stationary near A for s in S′ ⊂ S if there exists a neighborhood U of A for which h_{s,t} = h_{s,0} on U, for t in [0, 1], and for s in S′.

Similar definitions apply to sections of E and to solutions of Ω, with one restriction : families f : M × S → E of C^{k} sections of E are required to be C^{k}-continuous, that is, their k-jet extension j^{k}f : M × S → E^{k} is required to be a continuous map. To obtain solutions that are of smoothness class C^{r}, with k ≤ r ≤ ∞, one should everywhere consider C^{r} sections of E and C^{k−r} sections of E^{k} only. Also, when the relations considered is open, any (continuous) family of C^{r} sections of E can be approximated by a smooth family of C^{r} sections, that is a smooth map M × S → E.

In the sequel, S′ will always denote a subset of S = [0, 1]^{p} consisting of the union of some of its faces.

Definition 1.2 (The parametric h-principle.) The relation Ω ⊂ E^{k} is said to satisfy the parametric h-principle on M (near a subset A of M) if any family of sections of Ω (defined near A) is homotopic, among families of sections of Ω (defined near A), to a family of holonomic sections of Ω. If the sections are already holonomic on M (or near A) for s in S′, we may assume them to remain holonomic during the homotopy. (Equivalently, the homotopy may be assumed to be stationary for s in S′).

Remark 1.3 Observe that if a differential relation Ω satisfies the parametric h-principle, then its solutions satisfy some kind of uniqueness property. Indeed, let f_{0} and f_{1} be two solutions of Ω whose k-jet extensions are homotopic among sections of Ω. Let f_{t}^{1} be such a homotopy. Then the family f_{t}^{1} is homotopic to a family f_{t} of holonomic sections of Ω via a homotopy that is stationary for t ∈ {0, 1}. In particular, the two solutions f_{0} and f_{1} are homotopic among solutions of Ω.

Remark 1.4 The relation Ω satisfies the parametric h-principle on M if and only if the map j^{k} : Sol(Ω) → Γ(Ω) is a weak homotopy equivalence, where Sol(Ω) is the set of global solutions of Ω and where Γ(Ω) is the set of global sections of Ω (cf. [10] (C) p. 16).

1.2 Invariant relations

Definition 1.5 An isotopy of the manifold M is a family φ_{t}, t in [0, 1], of diffeomorphisms of M such that the map φ : M × [0, 1] → M : (x, t) ↦ φ_{t}(x) is smooth and φ_{0} = Id_{M}. Consider a foliation F on M. A foliated isotopy of (M, F) is an isotopy φ_{t} of M that preserves the foliation F, that is, (φ_{t})_{∗}(T F) = T F for all t in [0, 1]. Finally, two sets A and A′, with A′ ⊂ A are said to be isotopic (respectively foliated isotopic) if for every neighborhood U of A′, there exists an isotopy (respectively a foliated isotopy) φ_{t}^{U} of M such that

- φ_{t}^{U} is stationary near A′,
- φ_{t}^{U}(A) ⊂ A for all t,
- φ_{1}^{U}(A) ⊂ U.

Definition 1.6 (Invariant relations.) Let π : E → M be a locally trivial fibration. An isotopy φ_{t} of M is said to operate on local sections of E if we are given an isotopy \overline{φ}_{t} of E covering φ_{t} (i.e. π∘\overline{φ}_{t} = φ_{t} ∘π) satisfying the following property :

if φ_{t} coincides with φ_{t_{0}} on an open subset U for all t ≥ t_{0}, then \overline{φ}_{t}

coincides with \overline{φ}_{t_{0}} on π^{−1}(U) for all t ≥ t_{0}.

In this situation, if f : U → E|_{U} is a local section of E defined on some open subset U of M, then φ_{t} · f = \overline{φ}_{t}^{−1} ∘ f ∘ φ_{t} is a local section of E defined on φ_{t}^{−1}(U). This operation on local sections induces an operation on E^{k} as follows : φ_{t} · (j^{k}f(x)) = j^{k}(φ_{t} · f)(φ_{t}^{−1}(x)).

Definition 1.7 The differential relation Ω ⊂ E^{k} is said to be invariant under the isotopy φ_{t} if φ_{t} operates on local sections of E, and if Ω is invariant under the induced operation on E^{k} (i.e. φ_{t} · Ω = Ω). The relation Ω is said to be invariant (respectively foliated invariant) if it is invariant under all isotopies of M (respectively all foliated isotopies of (M, F)).

1.3 Local h-principle, h-principle for extensions

Definition 1.8 (The local parametric h-principle.) A differential relation is said to satisfy the local parametric h-principle if it satisfies the parametric hprinciple near any point.

Proposition 1.9 ([10] B_{2} p. 37) Any open differential relation satisfies the local parametric h-principle.

Definition 1.10 (The parametric h-principle for extensions.) Let (A, A′) be a nested pair of compact subsets of M. A differential relation Ω defined on M is said to satisfy the parametric h-principle for extensions of solutions form A′ to A, or on the pair (A, A′), if any family f_{s} of sections of Ω defined near A and holonomic near A′ is homotopic to a family of holonomic sections, through a homotopy that is stationary near A′. Moreover, if the sections f_{s} are already holonomic near A for s in S′, then the homotopy may be chosen to be stationary near A for s in S′.

Lemma 1.11 ([10]) Let Ω be a differential relation on the manifold M, and let (A, A′) be a nested pair of isotopic compact subsets of M. If Ω is invariant under all isotopies φ_{t}^{U}, where U runs through the set of neighborhoods of A′, then the parametric h-principle for extensions holds on the pair (A, A′).

Proposition 1.12 ([10] (A′) p. 40) Let Ω be a differential relation on M, and let M = ∪_{i≥0}K_{i} be an exhaustion of M by compact subsets (i.e. K_{i} ⊂ K_{i+1} and M = ∪_{i≥0}K_{i}). Suppose that Ω satisfies the parametric h-principle on K_{0}, as well as the parametric h-principle for extensions on all pairs (K_{i+1}, K_{i}). Then Ω satisfies the h-principle on M.

Consider an open differential relation Ω on a manifold M endowed with a foliation F.

Theorem 1.13 ([10] (B_{1}) p. 41, (C_{1}) p. 42, (C′_{3}) p. 43) Let (A, A′) be a nested pair of compact subsets of M such that the compact C = \overline{A − A′} is contained in an embedded submanifold M_{0} of M, of codimension at least one. If the relation Ω is invariant, then it satisfies the h-principle for extensions on (A, A′). If the relation Ω is only foliated invariant, but the submanifold M_{0} intersects the foliation F transversely, then Ω satisfies the h-principle for extensions on (A, A′) as well.

Remark 1.14 The hypothesis of the previous theorem can be weakened (without affecting the conclusion) as follows : C consists of a finite union of compact sets, each contained in an embedded submanifold of codimension at least one (transverse to F) (cf. [10] (A′) p. 40).

2 Outline of the proof

We begin this section with a rough outline of the proof of Theorem 0.1. Let Ω ⊂ E^{k} be an open, invariant differential relation on a manifold M. Fixing a section g of Ω, the procedure to deform g into a holonomic section of Ω is sketched below (the case of a family of sections is treated similarly). The hypotheses on Ω imply the following two facts.

(1) Since the relation Ω is open, it satisfies the local parametric h-principle (cf. Proposition 1.9). Thus, for any given point x in M, the section β can be deformed into a holonomic sections on a sufficiently small neighborhood of x.

(2) Since Ω is invariant, it satisfies the h-principle for extensions on any pair (A, A′) of isotopic compact subsets (cf. Lemma 1.11). Thus, if β can be deformed into a holonomic section near A′, it can also be deformed into a holonomic section near A.

Starting from a holonomic section ω of Ω defined on a neighborhood U of a point x and homotopic to β|_{U}, one tries to extend ω as far as possible. It can certainly be extended to any “large” open ball containing x (cf. (2)). But when trying to go further, one has to understand how to deal with the topology of M. A good grasp on the latter is provided by a proper, positive, Morse function on M, that is, a proper map f : M → [0, ∞) whose singular points are nondegenerate and lie on distinct levels. The term proper indicates that for any pair a < b of real numbers, the set f^{−1}([a, b]) is compact.

Let a < b be two noncritical values of f, and let ω be a holonomic section of Ω, defined on a neighborhood of f^{−1}([0, a]) and homotopic to β. Provided [a, b] does not contain any critical value of f, the set f^{−1}([0, b]) is isotopic to f^{−1}([0, a]) ([16]). The holonomic section ω can therefore be extended to a neighborhood of f^{−1}([0, b]) (cf. (2)). If [a, b] contains a critical value of f corresponding to a critical point x, the set f^{−1}([0, b]) is obtained from f^{−1}([0, a]) by gluing a closed disk D^{k} along its boundary, in the sense that f^{−1}([0, b]) is isotopic to f^{−1}([0, a])∪_{∂D^{k}} D^{k} ([16]). The dimension k of the disk coincides with the number of negative eigenvalues of the Hessian of f at x. Thus, to extend ω through x, one needs to be able to extend a holonomic section of Ω, homotopic to β, defined on a neighborhood of the boundary of an embedded disk D^{k}, to a holonomic section of Ω, homotopic to β, defined on a neighborhood of the entire disk. (In other words, one needs the relation Ω to satisfy the parametric h-principle for extensions on the pair (D^{k}, ∂D^{k})). This can be done as long as k < dim M (cf. Theorem 1.13), and constitutes the key step of the proof of Theorem 0.1.

The restriction k < dim M explains the dichotomy between closed and open manifolds. Indeed, if M is open, we may assume that f has no local maxima, or equivalently, that the disks we glue are never of maximal dimension. Beginning with a holonomic section ω of Ω defined near f^{−1}(0) and homotopic to β, we can therefore extend it to f^{−1}([0, b]) for larger and larger values of b, eventually obtaining a global holonomic section of Ω (cf. Proposition 1.12).

Introducing the gradient ∇f of f with respect to some Riemannian metric, one observes that its flow yields an isotopy between f^{−1}([0, b]) and f^{−1}([0, a]) ∪ ∂D^{k}D^{k}. Moreover, the embedded disk D^{k} may be thought of as the set of points in f^{−1}([a, b]) lying on trajectories θ(t) “converging” to x, in the sense that lim_{t→+∞} θ(t) = x. It is useful to adopt this point of view when dealing with foliated manifolds.

Consider now the corresponding problem in the foliated case. Let Ω be an open, foliated invariant differential relation defined on an open foliated manifold (M, F). Let f : M → [0, ∞) satisfy the hypotheses a), b) and c) of Definition 0.3. We may take advantage of what is already known. In particular, the facts (1) and (2) are still true here, provided we restrict ourselves to foliated isotopies. The key step is also valid in this context, provided the embedded disk D^{k}, does not only have codimension at least one, but intersects F transversely as well (cf. Theorem 1.13). The extra difficulties one faces come from the restriction to foliated isotopies. Indeed, up to foliated isotopy, the passage from f^{−1}([0, b]) to f^{−1}([0, b′]) does not correspond anymore to gluing some disk (nor even some family of disks). It corresponds instead to gluing the “skeleton” consisting of the set of bounded trajectories of the leafwise gradient vector field of f associated to some Riemannian metric g (cf. Definition 4.1 and Lemma 4.5). The skeleton can be very complicated due to two phenomena :

i) the leafwise critical points of f (Definition 3.2) can be leafwise degenerate (Remark 3.9),

ii) the leafwise critical points of f are not isolated in M (Remark 3.3).

Problem i) makes it hard to describe the topological structure of the skeleton already locally, near the foliated singular locus Σ_{f} (Definition 3.2). Fortunately, to apply the key step, we do not need to know the exact topological type of the skeleton, it suffices to know that it is contained in a finite union of compact subsets of submanifolds of codimension at least one, intersecting F transversely (cf. Remark 1.14). When the latter holds in a neighborhood of Σ_{f}, the metric g is said to be nice. The construction of a nice metric is detailed in Section 5. As it requires the foliated singular locus of f to be stratified according to Thom– Boardman (Subsection 3.1), the function f will be assumed to be strongly Fgeneric (Definition 3.22). As noticed in Remark 3.23, this extra assumption is not restrictive. The construction of the metric is done by successive extensions from a neighborhood of one stratum to the next (with respect to some natural order on the set of strata).

Problem ii), on the other hand, makes it hard to have a grasp on the global structure of the skeleton due to the presence of trajectories θ(t) for which both lim_{t→+∞} θ(t) and lim_{t→−∞} θ(t) are in f^{−1}([a, b]). For such a trajectory, the structure of the skeleton near lim_{t→−∞} θ(t) is quite complicated, more so that the stable and unstable manifolds of distinct critical points may not be assumed to intersect transversely. Fortunately, this difficulty vanishes if we approximate f by a function whose leafwise critical points are isolated with respect to the leaf topology (cf. Proposition 3.20), as it allows one to “slice” M sufficiently finely to (more or less) avoid having trajectories θ(t) for which both lim_{t→+∞} θ(t) and lim_{t→−∞} θ(t) lie in the same slice (cf. Section 6).

3 Genericity

This section is organized as follows. The first subsection shows how to adapt Boardman’s construction of a natural stratification of the singular locus of a smooth map ([3]) to the foliated case. We use Mather’s description of the Thom–Boardman stratification [14]. The second subsection shows that generically, the leafwise critical points of a real-valued function are isolated with respect to the leaf topology. This is a consequence of a result mentioned in [13] (6.1 p. 29) and proved along a scheme that appears in [12]. The third subsection defines (strongly) F-generic functions and exhibits some of their properties.

We begin with recalling the statement of Thom’s transversality theorem which plays a crucial role throughout this section, and with defining the terms leafwise critical point and foliated singular locus used in the previous section.

Given two manifolds M and N, the set C^{∞}(M, N) is endowed hereafter with the fine (or Whitney) C^{∞} topology. A subset of a topological space is residual if it is a countable intersection of dense open subsets. It is a classical result (e.g. [8]) that a residual subset of C^{∞}(M, N) is dense. Notice that a countable intersection of residual sets is still residual (while an intersection of dense sets is not dense in general). A condition on smooth functions f in C^{∞}(M, N) is said to be generic if it is satisfied by all functions in a residual subset of C^{∞}(M, N). Recall that a smooth map f : M → P intersects an embedded submanifold W of P transversely at a point x in M if and only if either f(x) ∉ W, or f(x) ∈ W and

f_{∗_{x}}T_{x}M + T_{f(x)}W = T_{f(x)}P ,  (1)

and that f intersects W transversely if (1) holds for all x in f^{−1}(W). In that situation, the set f^{−1}(W) is an embedded submanifold of M. If the map f is the inclusion of a submanifold W′, we say that the submanifolds W and W′ intersect transversely. Finally, the set of k-jets of local maps U ⊂ M → N is denoted by J^{k}(M, N).

Theorem 3.1 (Thom Transversality Theorem.) Let W be an embedded submanifold of J^{k}(M, N). The set of smooth maps f whose k-jet extension

j^{k}f : M → J^{k}(M, N)

is transverse to W is residual in C^{∞}(M, N).

A proof of the previous theorem can be found in [8].

Definition 3.2 Let (M, F) be a foliated manifold, and let f : M → ℝ be a smooth function.

- A point x at which d(f|_{F_{x}}) vanishes is called a leafwise critical point of f.

- The foliated singular locus of f (with respect to the foliation F) is the set of leafwise critical points of f, it is denoted by Σ_{f}.

Remark 3.3 An ordinary critical point of f is of course a leafwise critical point as well, but leafwise critical points persist in nearby leaves, so that they typically come in q-parameter families, where q is the codimension of F.

3.1 The Thom-Boardman stratification

Let E_{n} denote the set of germs of smooth maps ℝ^{n} → ℝ at the origin. The set E_{n} is a local ring, whose maximal ideal, denoted by 𝔪_{n}, is the set of germs vanishing at 0. Given an ideal I of the ring E_{n}, we use the notation I^{k} for the product I · . . . · I of k copies of I. Finally, the symbol D_{i} stands for the derivation \frac{∂}{∂x_{i}}, where x_{1}, …, x_{n} are the standard coordinates on ℝ^{n}.

Definition 3.4 Let I be a finitely generated ideal of E_{n}.

i) The rank of I, denoted by rk(I), is the dimension of (I + 𝔪_{n}^{2})/𝔪_{n}^{2} as a real vector space.

ii) The ideal generated by I and the set Γ_{r}(I) of r ×r minors of the Jacobian matrix (D_{i}f_{j})_{1≤i≤n,1≤j≤a}, where f_{1}, …, f_{a} is a set of generators for I, is denoted by Δ_{r}(I) and called a Jacobian extension of I. Notice that the ideal Δ_{r}(I) (unlike the set Γ_{r}(I)) does not depend on the choice of generators for I.

iii) The ideal Δ_{r+1}(I) with r = rk(I) is denoted by δ(I). Notice that when I is proper, the ideal δ(I) is proper as well.

iv) The Boardman symbol of I is the infinite sequence I(I) = (n − r_{1}, n − r_{2}, …, n − r_{ℓ}, …), where r_{ℓ} = rk(δ^{ℓ−1}(I)).

Let J^{k}(n, p) denote the set of k-jets of maps (ℝ^{n}, 0) → (ℝ^{p}, 0) at the origin. Given a jet z = j^{k}f(0) in J^{k}(n, p), represented by a map f, consider the ideal

I(z) =^{def.} (f) + 𝔪_{n}^{k+1} ,

where (f) denotes the ideal generated by the germs of the components f_{1}, …, f_{p} of f at 0. The Boardman symbol of the jet z, denoted by I(z), is defined to be the Boardman symbol of the ideal I(z) truncated at order k (notice that I(I(z)) is of the type (i_{1}, …, i_{k}, 0, …, 0, …)). Given a sequence I of k nonnegative integers, let Σ^{I} ⊂ J^{k}(n, p) denote the set of k-jets whose Boardman symbol is I.

Proposition 3.5 ([3]) Let I = (i_{1}, …, i_{k}). The set Σ^{I} is nonempty if and only if n ≥ i_{1} ≥ i_{2} ≥ … ≥ i_{k}, and either i_{1} > n − p, or i_{1} = n − p and i_{1} = i_{2} = … = i_{k}. Moreover Σ^{I} is an embedded submanifold of J^{k}(n, p) whose codimension is given by the formula

cod Σ^{I} = (p − n + i_{1})μ_{I} − (i_{1} − i_{2})μ_{sI} − … − (i_{k−1} − i_{k})μ_{s^{k−1}I} , (2)

where s^{j}I denotes the sequence (i_{j+1}, …, i_{k}), and where μ_{(i_{1},...,i_{ℓ})} is the number of sequences (j_{1}, j_{2}, …, j_{ℓ}) of integers satisfying

- j_{1} ≥ j_{2} ≥ … ≥ j_{ℓ},

- i_{r} ≥ j_{r} ≥ 0 for all r, and j_{1} > 0.

Given two manifolds N and P, of respective dimensions n and p, let J^{k}(N, P) denote the manifold of k-jets of local functions on N with values in P. The manifold J^{k}(N, P) is a bundle over N × P whose fibers are diffeomorphic to J^{k}(n, p), and whose structure group is the product of the group of invertible jets in J^{k}(n, n) with the group of invertible jets in J^{k}(p, p). For any sequence I = (i_{1}, …, i_{k}) of nonnegative integers, let Σ^{I} ⊂ J^{k}(N, P) denote the subset

Σ^{I} = ⋃_{(x,y)∈N×P} Σ_{x,y}^{I} ,

where Σ_{x,y}^{I} corresponds to Σ^{I} under the identification of the fiber of J^{k}(N, P) at (x, y) with the manifold J^{k}(n, p). Since Σ^{I} ⊂ J^{k}(n, p) is invariant under the action of the structure group of the bundle π : J^{k}(N, P) → N × P, the set Σ^{I} ⊂ J^{k}(N, P) is well-defined. Clearly Σ^{I} is an embedded submanifold of J^{k}(N, P) whose codimension is given by formula (2). Given a smooth function f : N → P whose k-jet extension intersects Σ^{I} nontrivially and transversely at x_{o}, let

Σ_{f}^{I} =^{def.} j^{k}f^{−1}(Σ^{I}) .

Σ_{f}^{I} is a submanifold of N in a neighborhood of x_{o}, and the following holds.

Proposition 3.6 ([3], Theorem (6.2))

dim Ker d(f|_{Σ_{f}^{(i_{1},...,i_{k})}})(x_{o}) = ℓ ⟺ j^{k+1}f(x_{o}) ∈ Σ^{(i_{1},...,i_{k},ℓ)} .

Let M be an m-dimensional manifold endowed with a foliation F of dimension n and codimension q. The inclusion of a leaf F into M gives rise to a submersion

r_{F} : π^{−1}(F × ℝ) ⊂ J^{k}(M, ℝ) → J^{k}(F, ℝ) : j^{k}f(x) ↦ j^{k}(f|_{F})(x) . (3)

For every sequence I = (n = i_{1}, …, i_{k}) of integers, let

Σ_{F}^{I} =^{def.} ⋃_{F ∈{leaves of F}} r_{F}^{−1} (Σ^{I}) .

The set Σ_{F}^{I} is an embedded submanifold of J^{k}(M, ℝ) whose codimension coincides with that of Σ^{I} in J^{k}(n, 1). Indeed, given a foliated chart (U, φ) centered at a point x, we have π^{−1}(U × ℝ) ∩ Σ_{F}^{I} ≃ U × ℝ × r^{−1}(Σ^{I}), where r is the submersion J^{k}(m, 1) → J^{k}(n, 1) : j^{k}f(0) ↦ j^{k}(f|_{ℝ^{n}×{0}})(0).

Definition 3.7 In C^{∞}(M) endowed with the fine C^{∞} topology, let A denote the set of functions whose k-jet extension is transverse to Σ_{F}^{I} for all I = (i_{1}, …, i_{k}) and all k = 1, 2, . . ..

By Thom’s Transversality Theorem, this set is residual. Now given f in A and any sequence I of length k, the set Σ_{F,f}^{I} = (j^{k}f)^{−1}(Σ_{F}^{I}) is an embedded submanifold whose codimension equals that of Σ_{F}^{I} in J^{k}(M, ℝ); it is called a Boardman stratum. There are only finitely many nonempty Boardman strata, provided we ignore sequences ending with more than one zero. The last assertion follows from the observation that if i_{k+1} > 0, the codimension of Σ^{(I,i_{k+1})} in J^{k+1}(n, p) is strictly larger than that of Σ^{I} in J^{k}(n, p) (cf. [3] p. 47).

The following result is a consequence of Proposition 3.6.

Proposition 3.8 Let f be an element of A and let x_{o} be a point in Σ_{F,f}^{(i_{1},...,i_{k})}. Then

dim (T_{x_{o}}F ∩ T_{x_{o}}Σ_{F,f}^{(i_{1},...,i_{k})}) = ℓ ⟺ j^{k+1}f(x_{o}) ∈ Σ_{F}^{(i_{1},...,i_{k},ℓ)} (4)

Proof. Since the problem is local, we may assume that M is diffeomorphic to a product F × V, with F = ℝ^{n}, V = ℝ^{q}, and where F corresponds to the “horizontal” foliation of F × V (whose leaves are the F × {v}’s). Let u_{1}, …, u_{n} and v_{1}, …, v_{q} denote the standard coordinates in F and V respectively. To the smooth function f : M → ℝ, we associate a smooth map f̃: M → ℝ × V by setting f̃(x) = (f(x), p(x)), where p denotes the natural projection F × V → V. The proof of Proposition 3.8 relies on the following three statements. Let x be any point in M, and let k be any positive integer, then

a) j^{k}f(x) ∈ Σ_{F}^{I} ⊂ J^{k}(M, ℝ) if and only if j^{k}f̃(x) ∈ Σ^{I} ⊂ J^{k}(M, ℝ × V);

b) j^{k}f intersects Σ_{F}^{I} transversely at x if and only if j^{k}f̃ intersects Σ^{I} transversely at x;

c) if x belongs to Σ_{F,f}^{I}, then T_{x}F ∩ T_{x}Σ_{F,f}^{I} = Ker d(f̃|_{Σ_{F,f}^{I}})(x).

Assuming these statements proved, we conclude the proof of Proposition 3.8 as follows. Statements a) and b), combined with Proposition 3.6 imply that

dim Ker d(f̃|_{Σ_{F,f}^{I}})(x_{o}) = ℓ ⟺ j^{k+1}f(x_{o}) ∈ Σ_{F}^{(i_{1},...,i_{k},ℓ)} .  (5)

Statement c) implies that (5) is equivalent to (4).

Proof of a). Consider the map

\begin{matrix}J^{k}(M, ℝ × V) & ⟶^{p_{1}×p_{2}} & J^{k}(M, ℝ) × J^{k}(M, V) \\ j^{k}g(x) & ↦ & (j^{k}g_{1}(x), j^{k}g_{2}(x)) , \end{matrix}

where g is a map M → ℝ × V, whose components are g_{1} : M → ℝ and g_{2} : M → V. Let P ⊂ J^{k}(M, V) be the image of the jet extension j^{k}p, and let P̃ = p_{2}^{−1}(P). It is sufficient to prove that in J^{k}(M, ℝ × V),

P̃ ∩ Σ^{I} = P̃ ∩ p_{1}^{−1}(Σ_{F}^{I}) .  (6)

Let z = (z_{1}, z_{2}) = (j^{k}f(x), j^{k}p(x)) = j^{k}f̃(x) be an element in P̃. We assume for convenience that x = 0. Let z̃_{1} denote j^{k}(f|_{F ×{0}})(x). We will prove, by induction on 0 ≤ ℓ ≤ k, that in E_{m},

δ^{ℓ}(I(z)) = (v) + δ^{ℓ}(I(z̃_{1})) ,  (7)

where (v) denotes the ideal generated by the germs of the functions v_{1}, …, v_{q}, and where δ^{ℓ}(I(z̃_{1})) is thought of as being an ideal of E_{m}. This will imply that rk δ^{ℓ}(I(z)) = q + rk δ^{ℓ}(I(z̃_{1})) for all 0 ≤ ℓ ≤ k, hence that z_{1} lies in Σ_{F}^{I} if and only if z lies in Σ^{I}, which is the content of (6).

Observe that (7) clearly holds when ℓ = 0. Indeed, in a neighborhood of 0 in M, the function f can be written as

f(u, v) = f(u, 0) + ∑_{i=1}^{q} v_{i}h_{i}(u, v)

for some smooth functions h_{i}. Then suppose that (7) holds for some ℓ ≥ 0. The ideal δ^{ℓ+1}(I(z)) is generated by δ^{ℓ}(I(z)) and the set Γ_{r+1}δ^{ℓ}(I(z)) of (r + 1) × (r + 1) minors of the Jacobian matrix (D_{i}f_{j})_{1≤i≤m,1≤j≤a}, where r = rk δ^{ℓ}(I(z)), and where f_{1}, …, f_{a} is a set of generators for δ^{ℓ}(I(z)). Since we may assume that f_{1} = v_{1}, …, f_{q} = v_{q} and that f_{q+1}, …, f_{a} are generators of δ^{ℓ}(I(z̃_{1})), the set Γ_{r+1}δ^{ℓ}(I(z)) is also the set of (r −q +1)×(r −q +1) minors of (D_{i}f_{j})_{1≤i≤n,q+1≤j≤a}. Since the latter set coincides with Γ_{r−q+1}δ^{ℓ}(I(z̃_{1})), we have

δ^{ℓ+1}(I(z)) = (v) + δ^{ℓ}(I(z̃_{1})) + Γ_{r−q+1}δ^{ℓ}(I(z̃_{1})) = (v) + δ^{ℓ+1}(I(z̃_{1})) .

Proof of b). Suppose that j^{k}f(x) ∈ Σ_{F}^{I}. Observe that the identity (6) implies that j^{k}f intersects Σ_{F}^{I} transversely at x if and only if

T_{z}P̃ = T_{z}P̃ ∩ T_{z} (p_{1}^{−1}Σ_{F}^{I}) + (j^{k}f̃)_{∗_{x}}(T_{x}M)
= T_{z} (P̃ ∩ p_{1}^{−1}Σ_{F}^{I}) + (j^{k}f̃)_{∗_{x}}(T_{x}M)
= T_{z} (P̃ ∩ Σ^{I}) + (j^{k}f̃)_{∗_{x}}(T_{x}M) ,

where z = j^{k}f̃(x) = (j^{k}f(x), j^{k}p(x)) = (z_{1}, z_{2}). So, in order to prove b), we need to show that Σ^{I} and P̃ intersect transversely at z, or equivalently that

T_{z_{2}}P + (p_{2})_{∗_{z}}(T_{z}Σ^{I}) = T_{z_{2}}J^{k}(M, V) .

Since J^{k}(M, V) ≃ M × V × J^{k}(m, q), and since P is transverse to {x} × V × J^{k}(m, q), it is sufficient to prove that T_{p(x)}V and T_{z_{2}}J^{k}(m, q) are contained in (p_{2})_{∗_{z}}(T_{z}Σ^{I}).

Observe that if φ_{X}^{t} denotes the local flow of a vector field X defined on M, vanishing at x, then j^{k}(f̃∘ φ_{X}^{t})(x) belongs to Σ^{I} for all t. Hence

ξ_{X} =^{def.} \frac{d (j^{k}(f̃∘ φ_{X}^{t})(x))}{dt}(0) ∈ T_{z}Σ^{I} .

Moreover, the set of vectors (p_{2})_{∗_{z}}(ξ_{X}), as X varies among vector fields vanishing at x, coincides with T_{z_{2}}J^{k}(m, q). So it remains to prove that T_{p(x)}V is contained in (p_{2})_{∗_{z}}(T_{z}Σ^{I}) as well. This is easily seen once we notice that T_{z}Σ^{I} contains T_{f̃(x)}(ℝ×V), and hence that (p_{2})_{∗_{z}}(T_{z}Σ^{I}) contains (p_{2})_{∗_{z}}(T_{f̃(x)}(ℝ×V)) = T_{p(x)}V.

Proof of c). From a) we know that Σ_{F,f}^{I} = ΣI ; f̃. Besides, if x lies in Σ_{F,f}^{I}, then

Ker d(f̃|_{ΣI ; f̃})(x) = Ker df̃(x) ∩ T_{x}ΣI ; f̃
= T_{x}F ∩ T_{x}ΣI ; f̃ .

Let f be an element of A. We would like to interpret the first few Σ_{F,f}^{I}’s. Before doing so, we introduce the foliated second differential of f (with respect to the foliation F), that is, the map

\overline{d}^{2}f : T_{Σ}F ⊕ T_{Σ}F → ℝ : (X, Y) → Y (X̃(f)) ,  (8)

where X̃ is any local section of T F extending X. As in the nonfoliated case, it is easily checked that \overline{d}^{2}f is well-defined and symmetric.

- Σ_{F,f}^{(n)} = Σ_{f} is the foliated singular locus of f (according to Definition 3.2).

- Σ_{F,f}^{(n,0)} is the set of leafwise nondegenerate critical points, i.e. the set of leafwise critical points x for which \overline{d}^{2}f(x) : T_{x}F ⊕ T_{x}F → ℝ is nondegenerate.

- Σ_{F,f}^{(n,i)} is the set of leafwise critical points x for which \overline{d}^{2}f(x) has rank n−i.

- In general, as implied by Proposition 3.8, Σ_{F,f}^{(i_{1},...,i_{k})} is the set of points x

in Σ_{F,f}^{(i_{1},...,i_{k−1})} where T_{x}F ∩ T_{x}Σ_{F,f}^{(i_{1},...,i_{k−1})} = i_{k}.

Observe that M is partitioned into the embedded submanifolds Σ_{F,f}^{I}, where I = (i_{1}, …, i_{k}, 0) and i_{k} ≠ 0. They intersect F transversely, in the sense that T_{x}Σ_{F,f}^{I} ∩T_{x}F = {0} for all x in Σ_{F,f}^{I}. Finally, the submanifold Σ_{F,f}^{I} is of course not closed in general. In fact,

\overline{Σ_{F,f}^{(i_{1},...,i_{k})}} ⊂ ⋃_{(j_{1},...,j_{ℓ})≥(i_{1},...,i_{k})} Σ_{F,f}^{(j_{1},...,j_{ℓ})} ,

where the symbol ≥ refers to the lexicographical order on the set of tuples of nonnegative integers.

Remark 3.9 A foliated manifold does not generally admit functions with no leafwise degenerate critical points. Indeed, for codimension one foliations, the foliated singular locus of such a function is a (not necessarily connected) closed transversal intersecting every compact leaf. The Reeb foliation on S^{3}, for example (cf. [4] p. 93), has no closed transversals intersecting the torus leaf (cf. [4] p. 147).

3.2 Isolatedness of leafwise critical points

We will need (cf. Observation 6.2) to approximate an F-generic function by one whose leafwise critical points are isolated with respect to the leaf topology. The purpose of this subsection is to prove that the latter property is generic (cf. Proposition 3.20).

Recall that E_{n} and 𝔪_{n} denote the ring of germs of maps ℝ^{n} → ℝ at the origin and its maximal ideal respectively. The set of k-jets of maps (ℝ^{n}, 0) → (ℝ, 0) at the origin is denoted by J^{k}(n). For ℓ > k, the natural map J^{ℓ}(n) → J^{k}(n) : j^{ℓ}f(0) → j^{k}f(0) is denoted by π_{k}^{ℓ}. As before, x_{1}, …, x_{n} are the standard coordinates on ℝ^{n}.

Definition 3.10 The Jacobian ideal of an element f ∈ 𝔪_{n}^{2}, denoted by J (f), is the ideal of E_{n} generated by the partial derivatives \frac{∂f}{∂x_{1}} , …, \frac{∂f}{∂x_{n}} of f. We say that f has codimension k if cod (J (f)) = dim_{ℝ} (𝔪_{n}/J (f)) = k.

We will think of a singular element z = j^{k}f(0) of J^{k}(n) (i.e. df(0) = 0) as a polynomial function on ℝ^{n}, and define its Jacobian ideal and codimension accordingly.

The following is a consequence of Nakayama’s lemma. A proof can be found in [12] (Appendix A.2., Proposition 2.2.).

Lemma 3.11 If f − f(0) ∈ 𝔪_{n}^{2} has finite codimension, then 0 is an isolated singularity of f.

Definition 3.12 Two germs f and g in E_{n} are said to be equivalent if there exists a germ ϕ of local diffeomorphism (ℝ^{n}, 0) → (ℝ^{n}, 0) of ℝ^{n} at the origin for which f = g ∘ ϕ.

Observe that equivalent germs have same codimension.

Definition 3.13 A germ f ∈ E_{n} is k-determined if for any element g in 𝔪_{n}^{k+1}, the germ f + g is equivalent to f.

Theorem 3.14 ([15]) If the codimension of f ∈ E_{n} is k, then f is (k + 2)- determined.

Consider in J^{k}(n) the set Z^{k} consisting of singular k-jets having codimension strictly larger than k − 2.

Proposition 3.15 The set Z^{k} is a real algebraic subset of the Euclidean space J^{k}(n).

Before proving Proposition 3.15, let us mention some facts about real algebraic sets. Recall that a real algebraic subset of a Euclidean space ℝ^{p} is the zero locus of a collection of polynomial functions defined on ℝ^{p}. Such a set is a variety when it is irreducible, that is, when it cannot be decomposed into the union of two proper real algebraic sets. A real algebraic subset S of ℝ^{p} can be written uniquely (up to reordering the factors) as a finite union

S = V_{1} ∪ … ∪ V_{n}

of varieties, where no factor V_{i} is contained in a factor V_{j} with i ≠ j. Such a decomposition is said to be minimal. Real algebraic sets can also be decomposed into smooth submanifolds :

Theorem 3.16 ([20]) Any real algebraic subset of ℝ^{p} can be decomposed into a finite union of disjoint embedded submanifolds of ℝ^{p}.

The dimension of S is defined to be the maximum dimension of the factors. The following result will be crucial later on (cf. proof of Proposition 3.19).

Theorem 3.17 ([20]) A proper subvariety of a variety has strictly smaller dimension.

Proof of Proposition 3.15. This type of argument is quite standard (see [12]). Let f ∈ J^{k}(n) be a singular jet. First observe that cod (J (f)) > k − 2 if and only if cod (J (f) + 𝔪_{n}^{k}) > k − 2. Indeed, suppose that cod (J (f) + 𝔪_{n}^{k}) ≤ k − 2. Then, in the following nested sequence of spaces

𝔪_{n} ⊃ J (f) + 𝔪_{n}^{2} ⊃ J (f) + 𝔪_{n}^{3} ⊃ … ⊃ J (f) + 𝔪_{n}^{k} ,

equality must occur somewhere, that is, J (f) + 𝔪_{n}^{r} = J (f) + 𝔪_{n}^{r+1} for some r ≤ k − 1. This implies that 𝔪_{n}^{r} ⊂ J (f) + 𝔪_{n}^{r+1}. Hence 𝔪_{n}^{k} ⊂ 𝔪_{n}^{r} ⊂ J (f) (cf. [13] Proposition 4. p. 3). Therefore cod (J (f)) = cod (J (f) + 𝔪_{n}^{k}) ≤ k − 2. The other implication is obvious.

Now cod (J (f) + 𝔪_{n}^{k}) > k − 2 is equivalent to

dim (J (f) + 𝔪_{n}^{k})/𝔪_{n}^{k} < dim (𝔪_{n}/𝔪_{n}^{k}) − (k − 2) =^{def.} N .

The vector space (J (f) + 𝔪_{n}^{k})/𝔪_{n}^{k} is generated by elements of the type x^{α} \frac{∂f}{∂x^{i}}, where α is a multi-index with |α| ≤ k − 2. Therefore, its dimension is less than N when the rank of a certain matrix M, whose coefficients are linear functions of the coefficients of the polynomial function f, is bounded by N − 1; or equivalently, when all minors of M of size N and larger vanish.

The combination of Theorem 3.15 and Theorem 3.16 implies that

Z^{k} = M_{1} ∐ … ∐ M_{r} ,

where each M_{i} is an embedded submanifold of J^{k}(n).

Proposition 3.18 For all ℓ > k, the real algebraic set Z^{ℓ} is a subset of the real algebraic set (π_{k}^{ℓ})^{−1}(Z^{k}).

Proof. Let f be an ℓ-jet whose truncation π_{k}^{ℓ}(f) does not belong to Z^{k}, that is, cod (J (π_{k}^{ℓ}(f))) ≤ k − 2. Theorem 3.14 implies that f is equivalent to π_{k}^{ℓ}(f). Hence cod (J (f)) ≤ k − 2 as well. Thus f lies outside Z^{ℓ}.

Proposition 3.19 The codimension c_{k} of Z^{k} in J^{k}(n) is an unbounded function of k.

Proof. Consider the homogeneous polynomial r = x_{1}^{k+1} + … + x_{n}^{k+1} of degree k + 1. The Jacobian ideal of r coincides with (x_{1}^{k}, …, x_{n}^{k}), and contains 𝔪_{n}^{n(k−1)+1}. Thus

cod (r) ≤ cod (𝔪_{n}^{n(k−1)+1}) =^{def.} a_{k} .

Let p be any polynomial of degree k. The codimension of p + r is bounded by the codimension of r (cf. [12] A.2., Theorem 2.7). Thus cod (p + r) ≤ a_{k} for all p ∈ J^{k}(n). Let ℓ = a_{k} + 2. We have just shown that for all p ∈ Z^{k}, the set (π_{k}^{ℓ})^{−1}(p) is not entirely contained in Z^{ℓ}.

Now, let Z^{k} = V_{1}^{k} ∪ … ∪ V_{n_{k}}^{k} be the minimal decomposition of Z^{k} into varieties. Let (π_{k}^{ℓ})^{−1}(V_{j}^{k}) be denoted by W_{j}. The real algebraic set W_{j} is irreducible as well. Because Z^{ℓ} = V_{1}^{ℓ} ∪ … ∪ V_{n_{ℓ}}^{ℓ} ⊂ (π_{k}^{ℓ})^{−1}(Z^{k}), the set V_{i}^{ℓ} coincides with (V_{i}^{ℓ} ∩ W_{1}) ∪ … ∪ (V_{i}^{ℓ} ∩ W_{n_{k}}). Since V_{i}^{ℓ} is irreducible, there exists an index j_{i} such that V_{i}^{ℓ} = V_{i}^{ℓ} ∩ W_{j_{i}}. The previous paragraph implies that V_{i}^{ℓ} is a proper subset of W_{j_{i}}. Hence, the codimension of V_{i}^{ℓ} is strictly larger than that of W_{j_{i}} (Theorem 3.17). Thus, the codimension of Z^{ℓ} in J^{ℓ}(n) is strictly larger than that of (π_{k}^{ℓ})^{−1}(Z^{k}), that is cod Z^{ℓ} > cod Z^{k}.

Proposition 3.20 Let M be a manifold endowed with a foliation F. For f in a residual subset B of C^{∞}(M), and for any leaf F of F, the critical points of f|_{F} are isolated in F.

Proof. Given a leaf F of the foliation F, define Z^{k}(F) ⊂ J^{k}(F, ℝ) to be

Z^{k}(F) = ⋃_{(x,y)∈F ×ℝ} Z^{k}(F)_{(x,y)} ,

where Z^{k}(F)_{(x,y)} corresponds to Z^{k} under the identification of the fiber at (x, y) of the bundle J^{k}(F, ℝ) → F × ℝ with J^{k}(n). Then, define Z^{k}(F) ⊂ J^{k}(M, ℝ) to be :

Z^{k}(F) = ⋃_{F ∈{leaves of F}} r_{F}^{−1}(Z^{k}(F)) ,

where r_{F} is defined by the expression (3). Locally, the set Z^{k}(F) is the union of finitely many embedded submanifolds of J^{k}(M, ℝ) having codimension at least c_{k}. Hence, Thom’s Transversality Theorem implies that, provided k is large enough for c_{k} > dim M, the set B of smooth real-valued functions on M whose k-jet does not meet Z^{k}(F) is residual in C^{∞}(M) with respect to the fine C^{∞} topology. Let f be such a function. For any leafwise singularity x of f, the jet j^{k}(f|_{F_{x}})(x) has codimension k − 2 at most. Thus, the germ of f|_{F_{x}} at x is equivalent to j^{k}(f|_{F_{x}})(x), and has finite codimension as well. Lemma 3.11 implies that the singularities of f|_{F_{x}} are isolated in F_{x}.

3.3 F-generic functions

Let M be an m-dimensional manifold carrying a foliation F of dimension n and codimension q.

Definition 3.21 (F-genericity.) A smooth real-valued function f on M is said to be F-generic if j^{1}f intersects Σ_{F}^{(n)} transversely and j^{2}f intersects Σ_{F}^{(i_{1},i_{2})} transversely for all pairs (i_{1}, i_{2}).

Definition 3.22 (Strong F-genericity). A smooth real-valued function on M is said to be strongly F-generic if

i) the critical points of f are nondegenerate and leafwise nondegenerate;

ii) for every sequence I of any length k, the map j^{k}f intersects Σ_{F}^{I} transversely;

iii) the leafwise critical points of f are isolated with respect to the leaf topology.

Here as well, it is a consequence of Thom’s Transversality Theorem that the set of (strongly) F-generic functions is residual in C^{∞}(M) for the fine C^{∞} topology. Indeed, the set of functions whose singularities are nondegenerate is a residual set (cf. [16]), as is the set of functions satisfying ii) (introduced in Definition 3.7), and the set of functions satisfying iii) (containing the set B introduced in Proposition 3.20). So the only thing that remains to be proven is that generically, critical points of f are leafwise nondegenerate. Introduce the following subsets of J^{2}(M, ℝ)

W^{ℓ} =^{def.} Σ^{(m,0)} ∩ Σ_{F}^{(n,ℓ)} 1 ≤ ℓ ≤ n .

One easily proves that the W^{ℓ}’s are embedded submanifolds of J^{2}(M, ℝ) of codimension strictly larger than m. Hence, the set of functions whose second jet does not meet any W^{ℓ} is residual. By construction, the critical points of those functions are leafwise nondegenerate.

Remark 3.23 Reflecting on Definition 0.3, one might think that c) is superfluous. Indeed, the set of strongly F-generic functions is dense in C^{∞}(M), and we can therefore approximate in the fine C^{∞} topology any given function by a strongly F-generic one. The problem is that one usually creates leafwise local maxima in the process. It is not clear, although quite plausible, that a function f with no leafwise local maxima always admits a nearby strongly F-generic function with no leafwise local maxima. However, if f is already F-generic, then any sufficiently (C^{∞}-) nearby strongly F-generic function will not have leafwise local maxima either.

The proof of the next lemma is very similar to that of the corresponding result in the single-leaf case (cf. [16] for example). Nevertheless, we include the proof for the sake of completeness.

Lemma 3.24 Let f : M → ℝ be a smooth function, and let x be a point in M for which j^{2}f(x) ∈ Σ_{F}^{(n,n−r)}. Then there exists a chart centered at x and adapted to F, with local coordinates x_{1}, …, x_{m}, with respect to which f has the following expression

f(x_{1}, … x_{m}) = ±x_{1}^{2} ± … ± x_{r}^{2} + f(0, …, 0, x_{r+1}, …, x_{m}) ,  (9)

Proof. Consider local coordinates x_{1}, …, x_{m} centered at x for which F is defined near x by the equations x_{n+1} = c_{n+1}, …, x_{m} = c_{m}, where c_{1}, …, c_{m} are constants . Since j^{2}f(x) ∈ Σ_{F}^{(n,n−r)}, the function f|_{F_{x}} is singular at x, and we may assume (after eventually performing a linear change of the coordinates x_{1}, …, x_{n}) that

(\frac{∂^{2}f}{∂x_{i}∂x_{j}} (0))_{1≤i,j≤r} = ( \begin{matrix}I_{d} & 0 \\ 0 & −I_{r−d} \end{matrix} ) .

One deduces from the implicit function theorem that there exists a neighborhood U of 0 in ℝ^{m−r}, a neighborhood U′ of 0 in ℝ^{r}, and a function α : (U, 0) → (U′, 0) such that, for every i = 1, …, r, and for x in U′ × U, one has

\frac{∂f}{∂x_{i}} (x_{1}, …, x_{m}) = 0 if and only if (x_{1}, …, x_{r}) = α(x_{r+1}, …, x_{m}) .

Use α to define new coordinates near x as follows :

\{ \begin{matrix}x′_{i} & = & x_{i} − α_{i}(x_{r+1}, …, x_{m}) & for & 1 ≤ i ≤ r \\ x′_{j} & = & x_{j} & for & j > r . \end{matrix}

Now

\frac{∂f}{∂x′_{i}} (0, …, 0, x′_{r+1}, …, x′_{m}) = 0 for 1 ≤ i ≤ r .

The primes are omitted hereafter. A classical argument (cf. [16] Lemma 2.1.) shows that in a convex neighborhood of 0, one has

f(x_{1}, …, x_{m}) = f(0, …, 0, x_{r+1}, …, x_{m}) + ∑_{1≤i≤j≤r} x_{i}x_{j}f_{ij}(x_{1}, …, x_{m}) . (10)

Consider the coordinates x′_{1}, …, x′_{m} defined near x by

\{ \begin{matrix}x′_{1} & = & x_{1}\sqrt{|f_{11}(x)|} + \frac{1}{2} ∑_{i=2}^{r} x_{i} \frac{f_{1i}(x)}{\sqrt{|f_{11}(x)|}} \\ x′_{i} & = & x_{i} for i ≥ 2 . \end{matrix}

With respect to these coordinates, f can be written as (10), where f_{11}(x) = 1, where f_{1i}(x) = 0, and where f_{ij}(0) remains unchanged. We repeat this process, modifying successively x_{2}, …, x_{r}, until the expression (9) is achieved. Observe that every change of coordinates that has been made preserves F since it leaves the coordinates x_{n+1}, …, x_{m} untouched.

Proposition 3.25 Let f : M → ℝ be an F-generic function without leafwise local maxima, and let x be in Σ_{F,f}^{(n,n−r)}. Then, for any system of coordinates centered at x satisfying the conclusion of the previous lemma, the quadratic form Q(x_{1}, …, x_{r}) = ±x_{1}^{2} ± … ± x_{r}^{2} occurring in expression (9) must have at least one positive sign.

Proof. Suppose on the contrary that either Q = 0 or Q is negative definite. We claim that, if so, any neighborhood of x contains leafwise local maxima of f, contradicting the assumption.

Let x ∈ M, and let (U, φ) be a chart at x, adapted to F, with local coordinates x_{1}, …, x_{m}, with respect to which f is expressed by (9). The chart (U, φ) induces a chart (Ũ, φ̃) on J^{2}(M, ℝ) :

\begin{aligned}φ̃ & : & Ũ = (π^{2})^{−1}(U) & → & φ(U) × ℝ × ℝ^{m} × 𝕊^{m} \\ & & p = j^{2}g(x) & ↦ & (φ(x), g(x), L_{p}, S_{p}) , \end{aligned}

where π^{2} denotes the natural projection J^{2}(M, ℝ) → M, and where 𝕊^{m} denotes the set of symmetric m × m matrices with real coefficients. The symbol L_{p} (respectively S_{p}) stands for the Jacobian (respectively Hessian) matrix of g ∘ φ^{−1} at φ(x).

It is convenient to decompose a symmetric n × n matrix S into blocks as follows :

S = ( \begin{matrix}A & B \\ B^{t} & D \end{matrix} ) ,

where A and D are r × r and (n − r) × (n − r) symmetric matrices, and where B is some r × (n − r) matrix. If A is nonsingular, then S has rank r if and only if the matrix e(S) = −B^{t}A^{−1}B + D vanishes. Define U to be the neighborhood of j^{2}f(x) in Ũ consisting of jets of local maps whose Hessian has a nonsingular upper left r × r block. Thus

Σ_{F,f}^{(n,n−r)} ∩ U = (ρ ∘ φ̃)^{−1}(0) ,

where ρ denotes the submersion :

\begin{aligned}ρ = ρ^{1} × ρ^{2} & : & φ̃(U) & → & ℝ^{n} × 𝕊^{n−r} \\ & & (x, t, L, S) & ↦ & (p(L), e(q(S))) , \end{aligned}

and where p : ℝ^{n} × ℝ^{m−n} → ℝ^{n} (respectively q : 𝕊^{m} → 𝕊^{n}) is the natural projection. Since j^{2}f (respectively j^{1}f) intersects Σ_{F}^{(n,n−r)} (respectively Σ_{f}) transversely at x, the maps ρ ∘ φ̃ ∘ j^{2}f and ρ^{1} ∘ φ̃ ∘ j^{2}f are submersions near x. Moreover (ρ^{1} ∘ φ̃ ∘ j^{2}f)_{∗_{x}}(T_{x}Σ_{f}) = 0, and thus

ρ^{2} ∘ φ̃ ∘ j^{2}f|_{Σ_{f} ∩U} : Σ_{f} ∩ U → 𝕊^{n−r}

is a submersion near x as well. This implies in particular that for ε sufficiently small, there exists a x_{o} in Σ_{f} ∩ U such that ρ^{2} ∘ φ̃ ∘ j^{2}f(x_{o}) = −εI_{n−r}. Then \overline{d}^{2}f(x_{o}) = −Id is negative definite. Thus f achieves a leafwise local maximum at x_{o}.

Remark 3.26 The assumption that f is F-generic and has no leafwise local maxima prevents f from having totally degenerate singularities (singularities at which the foliated second differential of f vanishes).

4 Leafwise gradient vector fields

Let (M, F) be a foliated manifold, let f be a positive, proper, strongly F-generic function on (M, F) (Definition 3.22), and let g be a Riemannian metric on M.

Definition 4.1 The leafwise gradient vector field of f associated to the metric g, denoted by ∇_{F}f, is the orthogonal projection onto T F of the (ordinary) gradient vector field of f. Equivalently, ∇_{F}f is the section of T F whose restriction to each leaf F is the gradient of f|_{F} associated to the metric g|_{F}.

The local flow of ∇_{F}f is denoted by φ^{t}. The orbit of a point x under the flow φ^{t} is denoted by θ_{x}. If S is some subset of ℝ, the portion {φ^{t}(x); t ∈ S} of θ_{x} is denoted by φ^{S}(x). Finally, the negative (respectively positive) limit set of the orbit θ_{x} is denoted by L_{x}^{−} (respectively L_{x}^{+}), i.e.

L_{x}^{−} = ⋂_{n=0}^{∞} \overline{φ^{(−∞,−n)}(x)} and L_{x}^{+} = ⋂_{n=0}^{∞} \overline{φ^{(n,∞)}(x)} .

Lemma 4.2

i) The set of points where ∇_{F}f vanishes coincides with Σ_{f}. Where ∇_{F}f does not vanish, (∇_{F}f)(f) > 0.

ii) For every x in M, the set L_{x}^{−} reduces to a single point, lying in Σ_{f}. Similarly, the set L_{x}^{+} is either empty if the trajectory of x is unbounded, or consists also of a single point, lying in Σ_{f} as well.

Proof.

i) Both assertions follow from the observation that for any X in T F, we have

Xf = df(X) = g(∇f, X) = g(∇_{F}f, X) .

ii) Since φ^{(−∞,0]}(x) is bounded (φ^{(−∞,0]}(x) ⊂ f^{−1}[0, f(x)]), for any sequence t_{k} → −∞ of real numbers, the sequence φ^{t_{k}}(x) has a converging subsequence . The set L_{x}^{−} is therefore nonempty. Moreover, because L_{x}^{−} is φ^{t}- invariant and contained in a level set of f (the one at level inf {f(φ^{t}(x)); t ∈ ℝ}), it must be contained in Σ_{f} (since f increases along nonconstant trajectories of ∇_{F}f).

Observe now that L_{x}^{−} is connected. Indeed, suppose it is not, and let O_{1} and O_{2} be disjoint open subsets of M containing L_{x}^{−} in their union. There exists a T ≤ 0 for which φ^{t}(x) ∈ O_{1} ∪ O_{2} as soon as t ≤ T. Otherwise there would be a sequence t_{k} → −∞ for which O_{1} ∪ O_{2} does not contain {φ^{t_{k}}(x)}, hence not either lim_{k→∞} φ^{t_{k}}(x). Connectedness of φ^{(−∞,T]}(x) implies that L_{x}^{−} is entirely contained in one of the O_{i}’s.

To conclude, we need to prove that L_{x}^{−} is contained in the leaf F_{x}. Suppose not, and let w be an point in L_{x}^{−} that does not belong to F_{x}. Let also (U, φ) be a chart adapted to F centered at w with φ(U) = [0, 1]^{n} × [0, 1]^{q}. Denote by P_{a} the plaque φ^{−1}([0, 1]^{n} ×{a}). Since f is strongly F-generic, its leafwise critical points are isolated for the leaf topology. We may therefore assume that w is the only point in Σ_{f} that lies in P_{0}. Because w belongs to L_{x}^{−} − F_{x}, there is a sequence a_{k} converging to 0 such that each plaque P_{a_{k}} contains a segment φ^{[t_{k},s_{k}]}(x) of θ_{x} that meets the boundary of P_{a_{k}} along its endpoints (i.e. φ^{t_{k}}(x) and φ^{s_{k}}(x) are in ∂P_{a_{k}}). We may also assume, without loss of generality, that t_{k} and s_{k} both converge to −∞. Now the sequence φ^{t_{k}}(x) has a subsequence that converges to a point w′ in ∂P_{0}. Since by construction w′ lies in L_{x}^{−}, it is necessarily a leafwise critical point of f, contradicting the hypothesis made earlier on U.

We have now all the ingredients needed : L_{x}^{−} is nonempty and connected, consists of leafwise critical points of f, and is entirely contained in a leaf. Property iii) of Definition 3.22 implies thus that L_{x}^{−} is a single point. The same argument proves the second part of statement ii) as well.

Definition 4.3 Given x in M, the limit point L_{x}^{−} is denoted by x^{−}. Similarly, when θ_{x} in bounded, the limit point L_{x}^{+} is denoted by x^{+}.

Definition 4.4

1. If A is a subset of Σ_{f}, we define the stable set of A to be the set W(A) = {x ∈ M ; x^{+} ∈ A}.

2. If A is a subset of M, the saturation of A is defined to be the set SA = {x ∈ M ; φ^{t}(x) ∈ A for some t ≥ 0}.

The following lemma will reduce the proof that the h-principle holds on M to the proof that it holds near the “skeleton” W(Σ_{f}). Let (a, b) be any pair of real numbers with a < b. The slice f^{−1}([a, b]) is denoted hereafter by M_{a}^{b}, and the set W(Σ_{f} ∩ M_{a}^{b}) ∩ M_{a}^{b} by W_{a}^{b}(Σ_{f}).

Lemma 4.5 Given any pair a < b of regular values of f, and given any neighborhood U of M_{0}^{a} ∪ W_{a}^{b}(Σ_{f}), the flow of ∇_{F}f yields a foliated isotopy ψ_{t} : M → M, t ∈ [0, 1], such that

- ψ_{t}(M_{0}^{b}) ⊂ M_{0}^{b} for all t in [0, 1],

- ψ_{1}(M_{0}^{b}) ⊂ U,

- ψ_{t} coincides with the identity map on M_{0}^{a} ∪ W_{a}^{b}(Σ_{f}),

- ψ_{t} is stationary for t ≥ \frac{1}{2}.

Proof. Let U be an open neighborhood of M_{0}^{a} ∪ W_{a}^{b}(Σ_{f}). As a first step, we prove existence of a neighborhood N of Σ_{f} ∩ M_{a}^{b} whose saturation SN is contained in U. Suppose that such an N does not exist. Consider the neighborhoods N_{k} of Σ_{f} ∩ M_{a}^{b} given by N_{k} = {x ∈ M; the distance between x and Σ_{f} ∩ M_{a}^{b} is less than \frac{1}{k} \}. Since Σ_{f} ∩ M_{a}^{b} is compact, N_{k} is in U for k large enough. Now, suppose that for every k there exists x_{k} in N_{k} such that for some negative t_{k}, the point y_{k} = φ^{t_{k}}(x_{k}) does not belong to U. There exists subsequences x_{k_{l}} and y_{k_{l}} of x_{k} and y_{k} respectively, converging to points x and y respectively. The point x lies in Σ_{f} ∩ M_{a}^{b}, and the point y lies outside U. Thus the trajectory θ_{y} intersects f^{−1}(b) − Σ_{f} nontrivially. Denote by H the foliation of M −Σ_{f} by the flow lines of ∇_{F}f. Let K be the portion of θ_{y} located between levels f(y) and level b. Since K is a compact subset of a leaf of H, there exists a closed neighborhood O of K in M − Σ_{f} such that H|_{O} is isomorphic to a product foliation. For l sufficiently large, y_{k_{l}} is contained in O, and therefore x_{k_{l}} and x are contained in O as well. This contradicts the hypothesis that O does not intersect Σ_{f}.

Now choose an open neighborhood N of Σ_{f} ∩M_{a}^{b} for which SN ⊂ U. Notice that SN is an open neighborhood of W_{a}^{b}(Σ_{f}). Since f^{−1}(a) is compact, there exists a δ > 0 such that f^{−1}([0, a + δ)) ⊂ U. Let Ũ = SN ∪ f^{−1}([0, a + δ)). Because f increases along nonconstant trajectories of ∇_{F}f, there exists, for every x in M_{a}^{b}, a t_{o} ≤ 0 such that φ^{t}(x) ∈ Ũ for all t ≤ t_{o}. Moreover, since M_{a}^{b} is compact, such a t_{o} can be chosen that does not depend on x. Hence φ^{t_{o}} shrinks M_{0}^{b} inside Ũ, but does not coincide with the identity on M_{0}^{a} ∪ W_{a}^{b}(Σ_{f}). Multiplying ∇_{F}f by a smooth function η : M → [0, 1] that equals 0 on M_{0}^{a} ∪ W_{a}^{b}(Σ_{f}) and 1 outside Ũ yields a vector field η∇_{F}f whose flow provides us with the desired isotopy. Indeed, let ψ_{t} = φ^{θ(t) t_{o}}, where θ is a smooth function [0, 1] → [0, 1] such that θ(0) = 0 and θ(t) = 1 for all t ≥ \frac{1}{2}. The isotopy ψ_{t} satisfies the four required properties.

5 Construction of the metric

Let M be an m-dimensional manifold endowed with a foliation F of dimension n and codimension q. Supposing the foliated manifold (M, F) open (Definition 0.3), let f : M → ℝ be a strongly F-generic function (Definition 3.22) without leafwise local maxima (cf. Remark 3.23). In this section, we carry out the construction of a Riemannian metric for which the stable set of the foliated singular locus of f (Definition 3.2 and Definition 4.4) is locally contained in a finite union of embedded submanifolds, of codimension at least one, transverse to the foliation F.

A submanifold S of M is said to intersect F transversely if for all s in S, the quantity dim (T_{s}S + T_{s}F) is maximal, that is, coincides with min{dim S + dim F, dim M}. Observe that a submanifold S whose dimension is strictly less that q does never intersect each leaf of F transversely. Nevertheless, it might intersect the foliation F transversely. Let g be a Riemannian metric on M. Recall the foliated second differential \overline{d}^{2}f of f (Subsection 3.1, formula (8)). Because the bilinear map \overline{d}^{2}f(σ) is symmetric for all σ ∈ Σ_{f}, the identity \overline{d}^{2}f(σ)(X, Y) = g(X, AY) defines a g-self-adjoint map A : T_{σ}F → T_{σ}F. The map A can be diagonalized by means of a g-orthogonal basis. We therefore have a splitting T_{σ}F = V_{σ}^{+} ⊕ V_{σ}^{−} ⊕ V_{σ}^{0} into positive, negative and null eigenspaces for A. The space V_{σ}^{0} is the kernel of \overline{d}^{2}f(σ) and is thus intrinsically defined (independently of g), unlike the other two. The distributions V^{±} : σ ∈ Σ_{f} ↦ V_{σ}^{±}, although not of constant rank, are smooth (unlike the distribution V^{0} : σ ∈ Σ_{f} ↦ V_{σ}^{0}). Smoothness for such a distribution is defined as follows : for every σ in Σ_{f} there exist local sections X_{1}, …, X_{d} of T M defined in a neighborhood U of σ in Σ_{f} such that

- {X_{1}(σ), …, X_{d}(σ)} is a basis of V_{σ}^{±},
- X_{i}(σ′) belongs to V_{σ′}^{±} for all σ′ in U.

Remark 5.1 Consider a sequence I = (i_{1}, …, i_{k−1}, i_{k} = 0). The rank of V^{0} and hence of V^{+} ⊕ V^{−} is constant on the submanifold Σ_{F,f}^{I} (cf. Section 3.1). Moreover, since both V^{+} and V^{−} are smooth distributions, their rank must be constant on any connected component of Σ_{F,f}^{I}.

Definition 5.2 The union of the connected components of Σ_{F,f}^{I} on which rk V^{+} achieves a fixed value d is denoted by Σ_{d}^{I}, and called hereafter a stratum of Σ_{f} (provided it is nonempty). We introduce an order on the set of strata as follows :

Σ_{e}^{(j_{1},...,j_{ℓ})} ≤ Σ_{d}^{(i_{1},...,i_{k})} ⟺ (n − e, j_{1}, …, j_{ℓ}) ≥ (n − d, i_{1}, …, i_{k}) ,

where ≥ refers to the lexicographical order on tuples of nonnegative integers.

Observe that because i_{k} = 0, a stratum S is an embedded submanifold (generally neither connected nor closed) that intersects F transversely, in the sense that T_{s}S ∩T_{s}F = {0} for all s in S. Moreover, there are only finitely many strata, and they partition Σ_{f}. Furthermore, the closure of Σ_{d}^{I} is contained in the union of the Σ_{e}^{J}’s with e ≤ d and J ≥ I (see end of Section 3.1). In particular, it is contained in the union of the smaller (with respect to the order introduced in Definition 5.2) strata. Consequently, the union of the l smallest strata is a closed set.

5.1 Outline of the construction of the metric

The construction is based on the following observation. Let σ be an element of Σ_{f}, and let x_{1}, …, x_{m} be local coordinates defined on a neighborhood U of σ in M with respect to which f has the following expression :

f(x_{1}, … x_{m}) = x_{1}^{2} + … + x_{d}^{2} + f(0, …, 0, x_{d+1}, …, x_{m}) ,

with d > 0 (Lemma 3.24 and Proposition 3.25). Let us introduce the following piece of notation : given an open subset O of M, let

W_{O}(Σ_{f}) = {x ∈ O ; x^{+} ∈ Σ_{f} ∩ O and φ^{t}(x) ∈ O ∀t ≥ 0} ,

where φ^{t} denotes the flow of the leafwise gradient vector field ∇_{F}f of f associated to the metric g (Definition 4.1), and where x^{+} = lim_{t→∞} φ^{t}(x) (cf. Definition 4.3). The set W_{O}(Σ_{f}) is the stable set (Definition 4.4) of Σ_{f} ∩ O with respect to (∇_{F}f)|_{O}. If instead of the metric g, we use the Euclidean metric with respect to the coordinates x_{1}, …, x_{m} to construct ∇_{F}f, then

W_{U}(Σ_{f}) ⊂ {x_{1} = 0, …, x_{d} = 0} ,  (11)

and the plane {x_{1} = 0, …, x_{d} = 0} is transverse to F, as needed.

The problem is that it is not clear that a global Riemannian metric g exists with the property that every point σ in Σ_{f} admits local coordinates as above, with respect to which g is Euclidean (at least along the leaves). Fortunately, we do not need that much : for the property (11) to hold, it is sufficient to have a Riemannian metric on U for which the plane {x_{1} = 0, …, x_{d} = 0} is orthogonal to the planes {x_{d+1} = c_{d+1}, …, x_{m} = c_{m}} (cf. Proposition 5.3). This constitutes the key observation, as it can be globalized, at least to a neighborhood of a stratum. Subsection 5.2 shows how. Briefly, starting with the (constant rank) distribution V^{+} on a stratum S, we construct a foliation G ⊂ F on a neighborhood N of S such that T_{S}G = V^{+} (the foliation G plays the role of the planes {x_{d+1} = c_{d+1}, …, x_{m} = c_{m}}). Then, we consider the manifold P of critical points of f along the leaves of G (the manifold P plays the role of the plane {x_{1} = 0, …, x_{d} = 0}). It is shown that any metric for which T P is orthogonal to T_{P} G has the property that W_{N} (Σ_{f}) ⊂ P (after eventually reducing N slightly).

Going from one stratum to the entire Σ_{f} is done by induction on the order of the strata (Subsection 5.3). Basically, to carry out this induction, we need to construct the foliation G in such a way that it “matches” the foliations already constructed near smaller order strata, meaning that where two such foliations coexist, the largest dimensional one contains the other one. The ordering is, roughly speaking, according to the number of strata that a given stratum contains in its closure. Closed strata are the smallest, strata having only closed strata in their closure come next, and so forth. This is dictated by the lexicographical ordering of the Boardman symbols, combined with the rank of the distribution V^{+}. Once a “coherent” set of foliations has been constructed, one may build a metric for which W_{N} (Σ_{f}) ⊂ P for all strata S (cf. Subsection 5.4).

5.2 The metric near one stratum

Let S be a (neither necessarily connected nor closed) embedded submanifold of Σ_{f}. Suppose that S intersects F transversely, and that the dimension of V^{+} is constant on S. Typically, S is a stratum of Σ_{f} (cf. Definition 5.2). Consider also exp : Õ ⊂ T M → M, the exponential map associated to g, defined on a fiberwise convex neighborhood Õ of the 0-section in T M. The construction of the metric near S is divided into the five following steps.

First step. The submanifold S is extended to an embedded submanifold S′ of M, transverse to F, and of dimension q = cod F.

If E = (T S + T_{S}F)^{⊥}, then exp |_{Õ∩E} is an embedding provided O = Õ ∩ E is sufficiently small. Define S′ = exp(O). Because exp_{∗} = Id along the 0-section, the bundle T_{S}S′ coincides with T S ⊕ E. In particular, the submanifold S′ is transverse to F along S. We may assume that O has been chosen small enough for S′ to be transverse to F everywhere.

Second step. Let D be a subbundle of T_{S}F (e.g. D = V^{+} when S is a stratum). We extend D to a subbundle D′ of T_{S′}F.

Let ∇ be a linear connection on T_{S′}F. For every x in S′, there is a natural path between x and a point x_{o} in S. Indeed, take γ_{x} : [0, 1] → S′ : t ↦ exp(tX), where exp(X) = x. The connection ∇ and the path γ_{x} induce a linear isomorphism i_{x} : T_{x_{o}}F → T_{x}F, obtained by parallel translation along γ_{x} with respect to ∇. For x in S′, define D′_{x} to be i_{x}(D_{x_{o}}).

Third step. Given an embedded submanifold S′, complementary to F, and a subbundle D′ of T_{S′}F, we extend D′ to a smooth foliation G, defined on a tubular neighborhood N of S′, tangent to F.

Consider the map

ξ : U ⊂ T_{S′}F → M : X_{s} ∈ T_{s}F ↦ τ_{X_{s}}(1) ,

where τ_{X_{s}} denotes the geodesic for the Riemannian metric g|_{F_{s}} starting at s and tangent to X_{s} (in particular τ_{X_{s}} ⊂ F_{s}), and where U is some fiberwise convex neighborhood of the 0-section on which ξ is an embedding. The map ξ is called hereafter the foliated exponential map associated to the metric g. Consider now the foliation G̃ of U by the traces of the cosets of D′, and the push-forward G of G̃ via the map ξ. Let N = ξ(U).

Fourth step. Suppose that the foliated second differential \overline{d}^{2}f of f is positive definite on D. Let P be the foliated singular locus of f with respect to the foliation G, that is, the set {x ∈ N ; df(T_{x}G) = 0}. Provided the neighborhood N of S is sufficiently small, the set P is an embedded submanifold of N transverse to G and closed in N.

Let p : BT G → N be the bundle of basis of T G, and define the map

η : BT G → ℝ^{d} : {e_{1}, …, e_{d}} ↦ (df(e_{1}), …, df(e_{d})) ,

where d is the dimension of G. Since \overline{d}^{2}f is nondegenerate on D = T_{S}G, the map η is a submersion near p^{−1}(S). Therefore, the map η is a submersion near η^{−1}(0), provided N is sufficiently small. The submanifold η^{−1}(0) coincides with p^{−1}(P), where P is a closed embedded submanifold of N. To prove that P is transverse to G, consider a vector X in T_{p}P ∩ T_{p}G, with p in P. Extend X to a section X̃ of T G defined on a neighborhood U of p in N. On P ∩U, the function X̃f vanishes, and because X is tangent to P, we have 0 = X(X̃f) = \overline{d}^{2}f(X, X) (notice that \overline{d}^{2}f is indeed well-defined on T_{P} G; it is the foliated second differential of f with respect to the foliation G). Since \overline{d}^{2}f is nondegenerate on T_{S}G = D, it remains nondegenerate on T_{P} G provided N is sufficiently small. Thus, the vector field X must vanish.

Fifth step. Let h be a Riemannian metric for which T P is orthogonal to T_{P} G. Such a metric can be constructed by means of a partition of unity for instance. The leafwise gradient vector field ∇_{F}f of f associated to h (Definition 4.1) satisfies the the following property.

Proposition 5.3 There exists a neighborhood M of N ∩ Σ_{f} contained in N for which

W_{M}(Σ_{f}) ⊂ P .

Proof. First observe that if x belongs to P, then (∇_{F}f)_{x} belongs to T P (as implied by the identity h(∇_{F}f, T_{x}G) = df(x)(T_{x}G) = 0). We will then show that for some sufficiently small neighborhood M of N ∩ Σ_{f}, no trajectory of (∇_{F}f)|_{M} starting outside P ∩ M can approach P ∩ M ⊃ Σ_{f} in forward time.

Since G ⊂ F, and since P is complementary to G, given x in P, there exist coordinates x_{1}, …, x_{m} on a neighborhood U of x such that

i) x_{n+1} = c_{n+1}, …, x_{m} = c_{m} define F|_{U},

ii) x_{d+1} = c_{d+1}, …, x_{m} = c_{m} define G|_{U}, iii) x_{1} = 0, …, x_{d} = 0 define P ∩ U,

iv) f(x_{1}, …, x_{m}) = x_{1}^{2} + … + x_{d}^{2} + f(0, …, 0, x_{d+1}, …, x_{m}).

The proof of this fact, very similar to that of Lemma 3.24, is omitted.

Let x be an element of P and let (x_{1}, …, x_{m}) be nice coordinates defined on a neighborhood U of x. Let also u = ∑_{i=1}^{d} u_{i} \frac{∂}{∂x_{i}} be a nonvanishing constant vector field on U tangent to the foliation G. In the next paragraph, a dot · will denote the Euclidean scalar product with respect to the coordinates (x_{1}, …, x_{m}). We have the following sequence of identities :

\frac{∂}{∂u}(∇_{F}f · u)(x) = ∑_{i,j=1}^{d} u_{i} \frac{∂}{∂x_{i}} (∑_{k=1}^{n} \frac{∂f}{∂x_{k}} h^{kj}u_{j}) (x)

= ∑_{i,j=1}^{d} ∑_{k=1}^{n} u_{i}u_{j} (\frac{∂^{2}f}{∂x_{i}∂x_{k}} (x)h^{kj}(x) + \frac{∂f}{∂x_{k}} (x)\frac{∂h^{kj}}{∂x_{i}} (x))

= 2h(u, u)(x) + ∑_{i,j=1}^{d} ∑_{k=1}^{n} u_{i}u_{j} \frac{∂f}{∂x_{k}} (x)\frac{∂h^{kj}}{∂x_{i}} (x) ,

where h^{kj} denotes the component k, j of the inverse of the matrix of h|_{T F×T F}. In particular, if x belongs to P ∩ Σ_{f} = N ∩ Σ_{f}, we have

\frac{∂}{∂u}(∇_{F}f · u)(x) = 2h(u, u)(x) > 0 .

Hence, for all x in P ∩ Σ_{f}, there exists an open neighborhood U_{x} of x, endowed with nice coordinates, such that \frac{∂}{∂u}(∇_{F}f · u) is strictly positive on U_{x}, for all nonvanishing constant vector field u tangent to the foliation G. As a consequence, a trajectory θ_{x} of ∇_{F}f for which θ_{x}(t) lies in U_{x} − (P ∩ U_{x}) for all t in [t_{o}, ∞) cannot converge to a point in P ∩ U_{x}. Indeed, Let w be any point in U_{x} − (P ∩ U_{x}), and let u be a nonvanishing constant vector field parallel to the line joining w = (x_{1}, …, x_{m}) to w_{o} = (0, …, 0, x_{d+1}, …, x_{m}). Then, since \frac{∂}{∂u}(∇_{F}f · u)|_{U_{x}} > 0 and (∇_{F}f · u)(w_{o}) = 0, the quantity (∇_{F}f · u)(w) is strictly positive. Supposing w = θ_{x}(t) for some t ≥ t_{o}, the last assertion implies that the Euclidean distance between θ_{x}(t) and P ∩ U_{x} grows with t.

Let {U_{i}; i ≥ 1} be a locally finite refinement of the covering {U_{x}; x ∈ P ∩Σ_{f}} of P ∩ Σ_{f}. Let M = N ∩ (∪_{i≥1}U_{i}). Any trajectory of (∇_{F}f)|_{M} starting in P ∩M remains in P ∩M, and no trajectory starting outside P ∩M will approach P ∩M in forward time. Indeed, suppose on the contrary that lim_{t→∞} θ_{x}(t) ∈ P, for some trajectory θ_{x} of (∇_{F}f)|_{M}. Then, for some t_{o} and some i, the point θ_{x}(t) lies in U_{i} for all t ≥ t_{0}. The discussion in the previous paragraph implies that θ_{x}(t) ∈ P for all t ≥ t_{0}, and hence that θ_{x} ⊂ P.

Remark 5.4 The reason for introducing M instead of supposing once more N small enough is that we need to make sure that the portion of Σ_{f} whose stable set is taken care of (N ∩ Σ_{f} in the previous proposition) is fixed once G is given.

Conclusion 5.5 Suppose given the following data :

- an embedded submanifold S of Σ_{f} intersecting F transversely,

- a subbundle D of T_{S}F restricted to which \overline{d}^{2}f is positive definite.

We can then construct a foliation G tangent to F, defined on a tubular neighborhood N of S, satisfying the following properties.

i) T_{S}G coincides with D.

ii) The foliated singular locus P of f with respect to the foliation G, is an embedded submanifold transverse to G (equivalently \overline{d}^{2}f is positive definite on T_{P} G). In particular, P as codimension at least one and intersects F transversely.

iii) Let h be a Riemannian metric on N for which T P is perpendicular to T_{P} G. Then there exists a neighborhood M of N ∩ Σ_{f} contained in N such that the leafwise gradient vector field of f associated to h satisfies the property that W_{M}(Σ_{f}) ⊂ P.

5.3 Construction of a complete system of foliations

As explained in the beginning of Section 5, the construction of a suitable Riemannian metric goes by defining a collection of foliations, one near each stratum, in a compatible way. The word compatible means that where two such foliations coexist, the one whose dimension is largest contains the other one. This construction is the subject of the present subsection.

Definition 5.6 Consider a collection of strata S_{1}, …, S_{l} satisfying the property that for all k, the set S_{k} ∪ … ∪ S_{l} is closed in M. A system of foliations on ∪_{k=1}^{l}S_{k} is a collection {G_{k}; k = 1, …, l} of foliations, where G_{k} is defined on a tubular neighborhood N_{k} of an open subset O_{k} = N_{k} ∩ S_{k} of S_{k}. Moreover, the following properties are required to hold :

a) For all 1 ≤ k ≤ l, S_{k} ⊂ N_{k} ∪ … ∪ N_{l}.

b) If S_{k_{1}} and S_{k_{2}} are such that \overline{S}_{k_{1}} ∩S_{k_{2}} = ∅ = S_{k_{1}} ∩\overline{S}_{k_{2}} then N_{k_{1}} ∩N_{k_{2}} = ∅.

c) dim G_{k} = rk (V^{+}|_{S_{k}}).

d) If dim G_{k_{1}} ≥ dim G_{k_{2}}, then T G_{k_{1}} ⊃ T G_{k_{2}} on N_{k_{1}} ∩ N_{k_{2}}.

e) For all k, the set P_{k} = {p ∈ N_{k}; df(p)(T_{p}G_{k}) = 0} is a submanifold of N_{k} transverse to G_{k}

A complete system of foliations on Σ_{f} is a system of foliations on the union of all the strata constituting Σ_{f}.

Remark 5.7 Notice that the set S_{1}, …, S_{l} of l smallest strata, arranged in decreasing order, satisfies the property that S_{k}∪…∪S_{l} is closed for all 1 ≤ k ≤ l.

Remark 5.8 Given a system of foliations on ∪_{k=1}^{l}S_{k}, we will always assume (and this is not restrictive) that another system of tubular neighborhoods \underline{N}_{k} of open subsets \underline{O}_{k} = \underline{N}_{k} ∩ S_{k} of S_{k} as been given that satisfies \overline{\underline{N}_{k}} ⊂ N_{k} for all k, as well as property a) in Definition 5.6. Hence, the collection \{G_{k}|_{\underline{N}_{k}}\} is also a system of foliations on ∪_{k=1}^{l}S_{k}. We will also use N ′_{k}, N ′′_{k}, …, N _{k}^{i}′, … to denote more tubular neighborhoods of open subsets O′_{k}, O′′_{k}, …, O_{k}^{i}′, … of S_{k} such that

\overline{\underline{N}_{k}} ⊂ N _{k}^{i}′ ⊂ \overline{N _{k}^{i}′} ⊂ N _{k}^{(i−1)}′ ⊂ N_{k} .

The need for these additional neighborhoods will appear in the proof of Proposition 5.9. It explains why, in Definition 5.6, we consider a tubular neighborhood of an open subset of a stratum, rather than a tubular neighborhood of the entire stratum. Indeed, if N is a tubular neighborhood of a nonclosed embedded submanifold S (like most strata), there is no tubular neighborhood N′ of S with \overline{N}′ ⊂ N. To obtain such an inclusion we have to replace S by a submanifold O of S with \overline{O} ⊂ S.

Proposition 5.9 (Recurrence step.) Let S_{1}, …, S_{l} be a set of strata satisfying the property that for all k, the set S_{k} ∪ … ∪ S_{l} is closed in M, and suppose that {G_{k}; k = 1, …, l} is a system of foliations on ∪_{k}S_{k}. Let S be another stratum for which S ∪ S_{1} ∪ … ∪ S_{l} is closed. Then we can extend {G_{k}; k = 1, …, l} to a system of foliations on S ∪ S_{1} ∪ … ∪ S_{l}.

The word extend has to be given the following meaning. A foliation G will be constructed on some tubular neighborhood N of an open subset O of S in such a way that G together with the restrictions of the foliations G_{k} to the neighborhoods \underline{N}_{k} is a system of foliations on S ∪ S_{1} ∪ … ∪ S_{l}.

Proof. We will assume for our convenience that \overline{S}∩S_{k} ≠ ∅ for all 1 ≤ k ≤ l (if this is not true, select the S_{k}’s intersecting \overline{S} nontrivially, label them S_{1}, …, S_{l}, and ignore the other ones until further notice). We can also assume that dim G_{k} ≥ dim G_{k+1} for all 1 ≤ k ≤ l − 1 (without affecting the property that S_{k} ∪ … ∪ S_{l} is closed for all k).

The proof follows the first three steps of Subsection 5.2. The difficulty lies in the second and third steps. We will need to adjust the distribution D′ and then the foliation G so as to make them match the foliations G_{k} already constructed.

Adjustment of D′. Suppose S ⊂ Σ_{d_{o}}^{(i_{1},...,i_{k})}, in particular rk V_{S}^{+} = d_{o}. We begin with describing a procedure that allows one to canonically extend a ddimensional subspace of T_{s}F, s ∈ S, on which \overline{d}^{2}f is positive definite (d may therefore not be greater than d_{o}), to a d_{o}-dimensional subspace with the same property.

Because \overline{d}^{2}f induces a metric on T_{S}F whose rank and signature are constant, it yields a bundle decomposition T_{S}F = V^{+}⊕V^{−}⊕V^{0} into g-orthogonal and \overline{d}^{2}forthogonal subspaces (cf. beginning of Section 5). Let p_{1} denote the projection T_{S}F → V^{+}, and let p_{2} denote the projection T_{S}F → V^{−}⊕V^{0}. Given an element X in T_{S}F, write X = X^{+} + X^{−} + X^{0}, where X^{+}, X^{−} and X^{0} belong to V^{+}, V^{−} and V^{0} respectively. For 1 ≤ d ≤ d_{o}, consider

W^{+} = ∪_{s∈S} (W_{s}^{+} = \{X ∈ T_{s}F ; \overline{d}^{2}f(X, X) > 0\})
P^{+,d} = ∪_{s∈S} (P_{s}^{+,d} = {P ∈ G^{d}(T_{s}F) ; P − {0} ⊂ W_{s}^{+}}) ,

where G^{d}(T_{s}F) denotes the Grassmann manifold of d-planes in T_{s}F. If E is a vector space endowed with an inner product, let S(E) denotes the unit sphere in E, and let B(E) denotes the open unit ball in E. Observe that if P is a plane in P^{+,d} then p_{1}|_{P} is injective. Hence, an element P of P_{s}^{+,d} is the graph of a linear map defined on p_{1}(P) :

\begin{aligned}φ_{P} & : & p_{1}(P) & → & V_{s}^{−} × V_{s}^{0} \\ & & x & ↦ & p_{2} ∘ (p_{1}|_{P})^{−1}(x) , \end{aligned}

whose restriction to the sphere S(p_{1}(P)) takes its values in B(V_{s}^{−}) × V_{s}^{0}. Conversely, an element of G^{d}(T_{s}F) that coincides with the graph of such a linear map belongs to P_{s}^{+,d}. Notice that if P is d_{o}-dimensional, then p_{1}(P) = V_{s}^{+}.

Now given an element P in some P_{s}^{+,d} with d < d_{o}, we extend it to an element E(P) in P_{s}^{+,d_{o}} as described hereafter. Let q_{P} : V_{s}^{+} → p_{1}(P) denote the orthogonal projection onto p_{1}(P). Then define E(P) via its associated linear map φ_{E(P)} by :

\begin{aligned}φ_{E(P)} & : & V_{s}^{+} & → & V_{s}^{−} × V_{s}^{0} \\ & & x & ↦ & (φ_{P} ∘ q_{P})(x) . \end{aligned}

Remark 5.10 We can form convex linear combinations of elements in P^{+,d_{0}}. Indeed, let P_{1}, …, P_{r} be elements of P_{s}^{+,d_{0}} for some s in S, and let a_{1}, …, a_{r} be positive real numbers with a_{1} + … + a_{r} = 1. Then a_{1}P_{1} + … + a_{r}P_{r} denotes the element of P_{s}^{+,d_{0}} whose associated function φ is a_{1}φ_{P_{1}} + … + a_{r}φ_{P_{r}}. The map φ takes its values in B(V_{s}^{−}) × V_{s}^{0}, as this set is convex.

As in the first step of Section 5.2, the stratum S is extended to a q-dimensio-nal embedded submanifold S′ transverse to F. The manifold S′ is the image of a fiberwise convex neighborhood O of the 0-section in the bundle p : E = (T S + T_{S}F)^{⊥} → S via the exponential map associated to g. Recall from the second step of Section 5.2, that a linear connection on T F determines, for each point x in S′, a linear isomorphism i_{x} : T_{x_{o}}F → T_{x}F, where x_{0} = exp(p(exp^{−1}(x))). Set O′_{k} = exp(O ∩ p^{−1}(N ′_{k} ∩ S)) and O′′_{k} = exp(O ∩ p^{−1}(N ′′_{k} ∩ S)).

Lemma 5.11 Provided O is sufficiently small, the following properties hold for all k :

- O′_{k} ⊂ N_{k},

- O′′_{k} ⊃ N ′′′_{k} ∩ S′,

- for every x in O′_{k}, the space i_{x}^{−1}(T_{x}G_{k}) is contained in W_{x_{o}}^{+} ∪ {0}.

Proof. Given a subset U of S and a positive number ε, let U^{ε} denote the subset exp(∪_{s∈U}B_{ε}(0_{s})) of S′, where B_{ε}(0_{s}) is the ball of radius ε centered at 0 in p^{−1}(s). For every s in S, there exists a neighborhood U_{s} of s in S and a positive number ε_{s} such that

- U_{s}^{ε_{s}} ⊂ N_{k} for all k for which s ∈ \overline{N} ′_{k},
- U_{s} ∩ N ′_{k} = ∅ for all k for which s ∉ \overline{N} ′_{k},
- U_{s} ⊂ N ′′_{k} for all k for which s ∈ \overline{N} ′′′_{k},
- U_{s}^{ε_{s}} ∩ N ′′′_{k} = ∅ for all k for which s ∉ \overline{N} ′′′_{k},
- i_{x}^{−1}(T_{x}G_{k}) ⊂ W_{x_{0}}^{+} ∪ \{0\} for all x ∈ U_{s}^{ε_{s}} with s in \overline{N} ′_{k}.

Let {U_{i}; i ≥ 1} be a locally finite refinement of the covering of S by the U_{s}^{ε_{s}}’s. For any fiberwise convex neighborhood O of the 0-section in E such that exp(O) ⊂ ∪_{i}U_{i}, the required three properties are satisfied.

For all k = 1, …, l, the dimension of G_{k} is at most equal to d_{o} (we supposed that \overline{S} ∩ S_{k} ≠ ∅ for all k, and this implies that rk V_{S}^{+} ≥ rk V_{S_{k}}^{+} for all k). We can therefore associate to each foliation G_{k} a d_{o}-dimensional distribution D′_{k}, defined on O′_{k}, tangent to F, and containing T G_{k}. Simply let

(D′_{k})_{x} = i_{x} (E (i_{x}^{−1}(T_{x}G_{k}))) .

We would like to paste the D′_{k}’s together so as to obtain a d_{o}-dimensional distribution D′ ⊂ T′_{S}F on S′ containing the distribution T_{S′∩N ′′′_{k}}G_{k}, and for which \overline{d}^{2}f is positive definite on D = D′|_{S}. For each k between 1 and l, let ρ_{k} : M → [0, 1] be a smooth function equal to 1 near N ′′_{k} and vanishing near the complement of N ′_{k}. We define D′ as follows :

D′_{x} = ρ_{1}(x_{o})(D′_{1})_{x} + (1 − ρ_{1}(x_{o}))ρ_{2}(x_{o})(D′_{2})_{x} + … +
(1 − ρ_{1}(x_{o})) … (1 − ρ_{k−1}(x_{o}))ρ_{k}(x_{o})(D′_{k})_{x} + … +
(1 − ρ_{1}(x_{o})) … (1 − ρ_{l}(x_{o}))i_{x}(V_{x_{o}}^{+}) ,

where x is in S′, and where x_{o} = exp(p(exp^{−1} x)). It is understood that when some distribution D′_{k} is not defined at the point x, the quantity ρ_{k}(x_{o})(D′_{k})_{x} is defined to be ρ_{k}(x_{o})i_{x}(V_{x_{o}}^{+}). The linear combinations appearing in the right hand side has to be understood as follows. If P_{1}, …, P_{r} are subspaces of T_{x}F for some x in S′ such that i_{x}^{−1}(P_{i}) ⊂ W^{+} ∪ {0}, and if a_{1}, …, a_{r} are real numbers, then

a_{1}P_{1} + … + a_{r}P_{r} = i_{x} (a_{1}i_{x}^{−1}(P_{1}) + … + a_{r}i_{x}^{−1}(P_{r})) ,

where the linear combination in the right hand side has to be interpreted according to Remark 5.10. Observe that on O′′_{k}, hence on N ′′′_{k} ∩ S′, the distribution D′ contains T G_{k}. Indeed, the function (1 − ρ_{k}(x_{o})) vanishes on N ′′_{k}, hence, only D′_{1}, …, D′_{k} are involved in the definition of D′_{x} for x in O′′_{k}. By construction, D′_{j} contains T G_{k} for j between 1 and k; hence, any convex linear combination of D′_{1}, …, D′_{k} contains T G_{k} as well.

To keep notations light, we denote N ′′′_{k} by N_{k}, N ′′′′_{k} by N ′_{k}, and so forth, while fixing \underline{N}_{k}.

Adjustment of G. We will now extend the distribution D′ to a foliation G ⊂ F defined on a tubular neighborhood N of S′, that contains G_{k} on N ∩N ′′′_{k}. Recall from the previous section the foliated exponential map :

ξ : U ⊂ T_{S′}F → M ,

where the set U is a fiberwise convex neighborhood of the 0-section in T_{S′}F, small enough for ξ to be an embedding. Let G̃_{k} denote the foliation (ξ)_{∗}^{−1}(G_{k}) defined on ξ^{−1}(N_{k}). It is of course tangent to the foliation of U by the fibers of the natural projection π : T_{S′}F → S′. For all k = 1, …, l, let E_{k} = (T G_{k})^{⊥} ∩ T F. Given any k, there exists a (not necessarily fiberwise convex) neighborhood V_{k} of the 0-section of T_{S′∩N_{k}}F contained in ξ^{−1}(N_{k}), such that each leaf of G̃_{k}|_{V_{k}} intersects E_{k} along exactly one point. In particular, there is a map g_{k} : V_{k} → E_{k} defined by {g_{k}(x)} = (G̃_{k})_{x}∩E_{k}, where (G̃_{k})_{x} denotes the leaf of G̃_{k}|_{V_{k}} containing x. The following lemma is very similar to Lemma 5.11 although it is formulated in T_{S′}F instead of M. Its proof is omitted. Let U′_{k} (respectively U′′_{k}) denote the set U ∩ π^{−1}(N ′_{k} ∩ S′) (respectively U ∩ π^{−1}(N ′′_{k} ∩ S′)).

Lemma 5.12 Provided U is sufficiently small, the following properties hold for all k.

- U′_{k} ⊂ V_{k},

- U′′_{k} ⊃ ξ^{−1}(N ′′′_{k}).

Let g_{l+1} denote the orthogonal projection T_{S′}F → D^{′⊥}, and let ν : T_{S′}F × T_{S′}F → T_{S′}F be the map that sends the pair (x, y) to the orthogonal projection of x onto the coset of D′ passing through the point y (that is, ν(x, y) = x + g_{l+1}(y − x)). Consider a smooth function ρ_{k} : M → [0, 1] whose value is 1 near N ′′_{k}, and is 0 near the complement of N ′_{k}. Define μ : U → T_{S′}F by

μ(x) = ν (x , ρ_{1}(x_{o})g_{1}(x) + (1 − ρ_{1}(x_{o}))ρ_{2}(x_{o})g_{2}(x) + … +
(1 − ρ_{1}(x_{o})) … (1 − ρ_{k−1}(x_{o}))ρ_{k}(x_{o})g_{k}(x) + … +
(1 − ρ_{1}(x_{o})) … (1 − ρ_{l}(x_{o}))g_{l+1}(x)) ,

where x_{o} = π(x), and where, if g_{k}(x) is not defined, we set ρ_{k}(x_{o})g_{k}(x) = 0_{x_{o}}. Observe that the map μ coincides with the identity on the 0-section. Denote by G′ the foliation of T_{S′}F by the cosets of D′. The idea is that, provided μ is a diffeomorphism, the foliation μ(G̃_{k}|_{U′′_{k}}) is tangent to the foliation G′.

Lemma 5.13 There exists a fiberwise convex neighborhood Ũ ⊂ U of the 0section in T_{S′}F for which μ : Ũ → T_{S′}F is a diffeomorphism onto its image, another neighborhood of the 0-section in T_{S′}F.

Proof. First observe that the map μ_{k}(x) = ν(x, g_{k}(x)), defined for x in V_{k}, coincides with the identity on V_{k} ∩ E_{k} (or on T_{S′}F for k = l + 1), and that (μ_{k})_{∗}(X) = X for X in T_{s}G_{k}, with s in S′ ∩ V_{k} (as a consequence of the fact that (D′)_{s} contains T_{s}G_{k} for s in S′ ∩ N_{k}). Hence, the map (μ_{k})_{∗_{s}} coincides with the identity map for s in S′ ∩ V_{k}. Observe now that, if we fix s in S′ and let x vary in T_{s}F ∩ U, the map μ can be written as :

μ(x) = ν(x, t_{1}g_{1}(x) + … + t_{l}g_{l}(x) + t_{l+1}g_{l+1}(x)) ,

where t_{k} = (1 − ρ_{1}(s)) … (1 − ρ_{k−1}(s))ρ_{k}(s) for 1 ≤ k ≤ l, and where t_{l+1} = (1 − ρ_{1}(s)) … (1 − ρ_{l}(s)). Hence, for X in T_{0_{s}}T_{s}F ≃ T_{s}F, we have

μ_{∗_{s}}(X) = ν_{∗_{(s,s)}} (X, (t_{1}g_{1} + … + t_{l+1}g_{l+1})_{∗_{s}} (X))
= ν_{∗_{(s,s)}} ((t_{1} + … + t_{l+1}) X, t_{1}(g_{1})_{∗_{s}}(X) + … + t_{l+1}(g_{l+1})_{∗_{s}}(X))
= t_{1} ν_{∗_{(s,s)}} (X, (g_{1})_{∗_{s}}(X)) + … + t_{l+1} ν_{∗_{(s,s)}} (X, (g_{l+1})_{∗_{s}}(X))
= t_{1} (μ_{1})_{∗_{s}}(X) + … + t_{l+1} (μ_{l+1})_{∗_{s}}(X)
= t_{1}X + … + t_{l+1}X
= X .

Thus μ_{∗_{s}} = Id for all s in S′, and the lemma follows.

Lemma 5.14 For all k, the foliation μ_{∗}^{−1}(G′) contains the foliation G̃_{k} on U′′_{k} ∩ Ũ, hence on ξ^{−1}(N ′′′_{k}) ∩ Ũ.

Proof. If x belongs to U′′_{k} ∩ Ũ, then (1 − ρ_{k}(x_{o})) = 0. Hence the map μ can be expressed as follows :

μ(x) = ν (x, ρ_{1}(x_{o})g_{1}(x) + (1 − ρ_{1}(x_{o}))ρ_{2}(x_{o})g_{2}(x) + … +
(1 − ρ_{1}(x_{o})) … (1 − ρ_{k−1}(x_{o}))ρ_{k}(x_{o})g_{k}(x)) .

Besides, for all j = 1, …, k, the map x ↦ ρ_{j}(x_{o})g_{j}(x) is constant on the leaf (G̃_{k})_{x} of G̃_{k}|_{U′′_{k}∩Ũ} passing through x. This assertion follows from the fact that whenever k_{1} > k_{2}, the distribution T G_{k_{1}} contains the distribution T G_{k_{2}} on N_{k_{1}} ∩ N_{k_{2}}. Thus the map μ sends the entire leaf (G̃_{k})_{x} into a coset of D′.

The foliation G is defined to be ξ_{∗}(μ_{∗}^{−1}(G′)|Ũ). Its domain is the open set N = ξ(Ũ). Lemma 5.14 implies that G contains G_{k} on N ′′′_{k} ∩ N. Define P to be the foliated singular locus of f with respect to G, that is, P = {p ∈ N; df(p)(T_{p}G)}. We may assume, after shrinking N if needed, that \overline{d}^{2}f is positive definite on T_{P} G (cf. Fourth step of Subsection 5.2). The strata we might have ignored in the very beginning of this proof should now be re-incorporated.

Lemma 5.15 The foliations G, G_{1}, …, G_{l}, defined on N , \underline{N}_{1}, …, \underline{N}_{l} respectively, form a system of foliations on S ∪ S_{1} ∪ … ∪ S_{l}.

Proof. Among the five defining properties of a system of foliations, properties a), c), d) and e) have been taken care of during the construction. Only b) requires some attention. We need to make sure that N does not intersect the tubular neighborhoods of the strata that we discarded at the beginning of the proof. A way to insure this is to fix a family of neighborhoods Ñ_{k}, one for each strata S_{k}, satisfying the following property :

if \overline{S}_{k_{1}} ∩ S_{k_{2}} = ∅ = S_{k_{1}} ∩ \overline{S}_{k_{2}}, then Ñ_{k_{1}} ∩ Ñ_{k_{2}} = ∅ as well.

Then, whenever we consider a neighborhood of some strata S_{k}, we request that it be contained in Ñ_{k}.

End of the proof of Proposition 5.9.

Corollary 5.16 A complete system of foliations on Σ_{f} exists.

Proof. Let S_{1}, …, S_{l} be the set of all strata of Σ_{f}, presented in such a way that S_{1} > S_{2} > …. Then, as described in Subsection 5.2 (or in Proposition 5.9 with l = 0), we can construct a foliation G_{l} near S_{l} and use Proposition 5.9 repeatedly to extend G_{l} to a system of foliations near ∪_{k}S_{k}.

5.4 Construction of a metric from a complete system of foliations

Let {G_{k}; k = 1, …, l} be a complete system of foliations on Σ_{f}. Each G_{k} is defined on a tubular neighborhood N_{k} of an open subset O_{k} of the stratum S_{k}. The foliated singular locus of f with respect to G_{k} is denoted by P_{k}. We suppose that dim G_{k} ≥ dim G_{k+1} for all k.

Definition 5.17 A Riemannian metric is said to be adapted to the system of foliations {G_{k}; k = 1, …, l} if T_{x}P_{k} is perpendicular to T_{x}G_{k} for all x in P_{k}∩N ′_{k}.

Proposition 5.18 A Riemannian metric adapted to the system of foliations {G_{k}; k = 1, …, l} exists.

Proof. For x in M, let U_{x} be a neighborhood of x in M, and let h_{x} be a Riemannian metric defined on U_{x} such that 1. U_{x} ⊂ N_{k} for all k for which x ∈ P_{k} ∩ \overline{N} ′_{k},

2. U_{x} ∩ P_{k} ∩ \overline{N} ′_{k} = ∅ for all k for which x ∉ P_{k} ∩ \overline{N} ′_{k},

3. if x ∈ P_{k} ∩ \overline{N} ′_{k}, then T_{y}P_{k} ⊥_{h_{x}} T_{y}G_{k} for all y in U_{x} ∩ P_{k}.

Existence of U_{x} and h_{x} is easily seen, except perhaps for the last condition. Suppose that x belongs to \overline{N} ′_{k} ∩ P_{k} if and only if k ∈ \{k_{1}, k_{2}, …, k_{r}\}, with k_{1} < k_{2} < … < k_{r}. Then for Property 3. to hold, it is sufficient that the following bundles be pairwise h_{x}-orthogonal.

T P_{k_{1}}, T P_{k_{2}} ∩ T G_{k_{1}}, T P_{k_{3}} ∩ T G_{k_{2}}, …, T P_{k_{r}} ∩ T G_{k_{r−1}}, T G_{k_{r}} .

These bundles span T M on U_{x} ∩ (∩_{j}(\overline{N} ′_{k_{j}} ∩ P_{k_{j}})) and are linearly independent, so that the Gram-Schmidt orthogonalization process can be carried out.

let {U_{i}; i ≥ 1} be a locally finite refinement of the covering of M by the U_{x}’s, and let {θ_{i}} be a partition of unity subordinate to the covering {U_{i}}. Denote by h_{i} the metric h_{x_{i}}, where x_{i} is chosen in such a way that U_{i} ⊂ U_{x_{i}}. Define

h = ∑_{i≥1} θ_{i} h_{i} .

Then T_{y}P_{k} ⊥_{h} T_{y}G_{k} whenever y belongs to P_{k} ∩ N ′_{k}. Indeed, let y ∈ P_{k} ∩ N ′_{k}, and suppose that y belongs to U_{i} ⊂ U_{x_{i}} for some i. Then x_{i} ∈ P_{k} ∩ \overline{N} ′_{k}. Hence T_{y}P_{k} is perpendicular to T_{y}G_{k} with respect to the metric h_{i}. Since this holds true for every i for which y ∈ U_{i}, we have T_{y}P_{k} ⊥_{h} T_{y}G_{k}.

5.5 Conclusion

The preceding subsections show how to construct a complete system of foliations {G_{k}; k = 1, …, l} on Σ_{f} (Definition 5.6 and Corollary 5.16), as well as a Riemannian metric h adapted to that system (Definition 5.17 and Proposition 5.18). By Proposition 5.3, there exists, for every k, a neighborhood M_{k} of N ′_{k} ∩ Σ_{f} for which

W_{M_{k}}(Σ_{f}) ⊂ P_{k} ,

where P_{k} denotes the foliated singular locus of f with respect to the foliation G_{k}. The submanifold P_{k} has codimension at least one and intersect F transversely. Notice that the M_{k}’s cover Σ_{f}.

6 The proof

Let us recall the statement of Theorem 0.4 whose proof will be completed in the present section.

Theorem 6.1 On an open foliated manifold, any open relation invariant under foliated isotopies satisfies the parametric h-principle.

Proof. Let Ω be an open relation invariant under foliated isotopies on the open foliated manifold (M, F). Consider a proper F-generic function f : M → [0, ∞), without leafwise local maxima. As observed in Remark 3.23, we may assume, without loss of generality, that f is not only F-generic but also strongly F-generic. Let Σ_{f} = S_{1} ∪ … ∪ S_{l} be the decomposition of the foliated singular locus of f (Definition 3.2) into strata (Definition 5.2). Let {G_{k}; k = 1, …, l} be a complete system of foliations (Definition 5.6 and Corollary 5.16), endowed with an adapted metric h (Definition 5.17 and Proposition 5.18). As before, we will denote by N_{k} the domain of definition of the foliation G_{k}, by P_{k} the foliated singular locus of f with respect to the foliation G_{k}, and by M_{k} a neighborhood of N_{k} ∩ Σ_{f} for which W_{M_{k}}(Σ_{f}) ⊂ P_{k}. The leafwise gradient vector field of f with respect to h (Definition 4.1) is denoted as before by ∇_{F}f, and its local flow by φ^{t}. We will need a partition P = { a_{0} = 0 = a_{1} < a_{2} < … < a_{i} < … } of [0, ∞) by non critical values (except for a_{0} and a_{1}) of f. The latter provides us with an exhaustion M = ∪_{i}K_{i} of M by compact subsets K_{i} = f^{−1}([a_{i}, a_{i+1}]).

The proof of Theorem 6.1 consists in showing that, provided the partition P is fine enough, the hypotheses of Proposition 1.12 are satisfied. The first hypothesis, that the h-principle is valid near K_{0} is easy to handle. Indeed, K_{0} = f^{−1}(0) is a finite union of points (we assume here that the minimum value of f is 0), and the local h-principle (Definition 1.8) is valid for any open relation (cf. Proposition 1.9), in particular for Ω. Concerning the second hypothesis, that the h-principle for extensions is valid on each pair (K_{i+1}, K_{i}), it is proven in two steps. The combination of Lemma 4.5 and Lemma 1.11 proves the h-principle for extensions on each pair (K_{i+1}, K_{i} ∪ W_{a_{i+1}}^{a_{i+2}}(Σ_{f})). It remains to prove that the h-principle for extensions is valid on each pair (K_{i} ∪ W_{a_{i+1}}^{a_{i+2}}(Σ_{f}), K_{i}). This is where the construction of a Riemannian metric carried out in Section 5 is needed, as will become clear below. Fix a slice K_{i}.

Observation 6.2 Since f is a strongly F-generic function on (M, F), for each leaf F of the foliation F, the critical points of f|_{F} are isolated in F. Moreover, the submanifold Σ_{f} is closed and embedded. Hence, for each σ in Σ_{f}, there exists a neighborhood U_{σ} of σ in M satisfying the following properties :

1. U_{σ} is the domain of a chart (U_{σ}, φ_{σ}) adapted to F and centered at σ such that φ_{σ}(U_{σ}) = B_{σ}^{1} ×B_{σ}^{2}, where B_{σ}^{1} is a closed ball about φ(σ) in the image of the leaf F_{σ}, and where B_{σ}^{2} is a closed ball about φ(σ) in the transverse direction.

2. U_{σ} is contained in M_{k} for some k.

3. The compact set bU_{σ} =^{def.} φ_{σ}^{−1}(∂B_{σ}^{1} × B_{σ}^{2}) does not intersect Σ_{f}.

Let {U_{r}; r = 1, 2, …} be a locally finite refinement of the cover of Σ_{f} by the U_{σ}’s. Observe that ∪_{r}bU_{r} is a closed set, and that for only finitely many r’s, the set U_{r} intersects K_{i} nontrivially. For each r, choose a σ_{r} in Σ_{f} such that U_{r} ⊂ U_{σ_{r}}. Choose also an index k_{r} ∈ {1, …, l} for which U_{r} ⊂ M_{k_{r}}. We will use the notations introduced in Section 4.

Lemma 6.3

W(Σ_{f}) ⊂ ⋃_{r} S(U_{r} ∩ P_{k_{r}}) .

Proof. Let x be a point in M − Σ_{f} such that x^{+} is in Σ_{f}. The critical point x^{+} belongs to U_{r} for some r. Either the trajectory φ^{[0,∞)}(x) intersects bU_{r}, or it is entirely contained in U_{r}. In the second case, we deduce from Proposition 5.3, and from the fact that U_{r} ⊂ M_{k_{r}}, that x must be contained in U_{r} ∩ P_{k_{r}}. In the first case, let x_{0} = φ^{t}(x) be the point in φ^{[0,∞)}(x)∩bU_{r} for which t is maximum. Then φ^{[t,∞)}(x) is entirely contained in U_{r}, and Proposition 5.3 implies that x_{0} must belong to P_{k_{r}}. In both cases, x belongs to S(U_{r} ∩ P_{k_{r}}).

Lemma 6.4 Let

ε_{i} = inf { f(x) − f(x^{−}) ; x ∈ ∪_{r}(bU_{r} ∩ P_{k_{r}}) ∩ K_{i} and x^{−} ∈ K_{i} } .

Then ε_{i} is positive.

Proof. Suppose on the contrary that ε_{i} = 0. Then there exists a sequence (x_{j}) of points in ∪_{r}bU_{r} ∩K_{i} with x_{j}^{−} in K_{i} such that lim_{j→∞} (f(x_{j}) − f(x_{j}^{−})) = 0. We may assume, after extracting a subsequence if necessary, that the sequence (x_{j}) converges in K_{i} to a point x. Since x belongs necessarily to ∪_{r}bU_{r}, it is not a leafwise critical point of f, and we may therefore consider a chart (U, ψ) about x, adapted to the foliation H of M − Σ_{f} by the orbits of ∇_{F}f. For all sufficiently large j, the point x_{j} lies in U. Letting x′ = φ^{t}(x) with t < 0 be an element in U, there exists a sequence x′_{j} = φ^{t_{j}}(x_{j}) with t_{j} < 0 converging to x′. Because f is strictly increasing along nonconstant trajectories of ∇_{F}f,

0 < f(x) − f(x′)
= lim_{j→∞} (f(x_{j}) − f(x′_{j}))
≤ lim_{j→∞} (f(x_{j}) − f(x_{j}^{−})) ,

contradicting the hypothesis that lim_{j→∞} (f(x_{j}) − f(x_{j}^{−})) = 0.

Now let P′ = {b_{0} = 0 = b_{1} < b_{2} < … < b_{j} …} be a refinement of the partition P such that, if a_{i} = b_{j_{i}}, then b_{j+1} − b_{j} < ε_{i}, whenever j_{i} ≤ j < j_{i+1}. Fix j, let L_{j} denote the slice f^{−1}([b_{j}, b_{j+1}]), and let i be the index for which [b_{j}, b_{j+1}] ⊂ [a_{i}, a_{i+1}].

Lemma 6.5 For all r, the set N_{r} = S(U_{r} ∩ P_{k_{r}} ∩ L_{j}) ∩ L_{j} is a finite union of compact subsets of embedded submanifolds of codimension at least one, transverse to F.

Remark 6.6 It is not true in general that N_{r} is an embedded submanifold. The problem is that, even though ∇_{F}f is a leafwise gradient vector field, there might be trajectories leaving U_{r}∩P_{k_{r}} and coming back later to U_{r}∩P_{k_{r}}, creating in the process self-intersections in N_{r}.

Proof of Lemma 6.5. As in the proof of Lemma 6.4, let H denote the foliation of M − Σ_{f} by the orbits of ∇_{F}f. For any point x in bU_{r} ∩ P_{k_{r}} ∩ L_{j}, there exists a t_{x} ≤ 0 for which φ^{t_{x}}(x) = x_{0} belongs to f^{−1}(b_{j}) − Σ_{f}. Indeed, b_{j+1} − b_{j} < ε_{i}, but f(x) − f(x^{−}) ≥ ε_{i}, by definition of ε_{i}. Let V_{x} be an open neighborhood of φ^{[t_{x},0]}(x) in M such that

- H|_{V_{x}} is isomorphic to a product foliation,

- each leaf of H|_{V_{x}} intersects f^{−1}(b_{j}) nontrivially.

Let also U_{x} be a neighborhood of x in V_{x} ∩ M_{k_{r}} for which U_{x} ∩ P_{k_{r}} ≃ U_{x}^{1} × U_{x}^{2}, where

- U_{x}^{1} is a neighborhood of x in the leaf H_{x} of H through x,

- U_{x}^{2} is an submanifold containing x whose projection onto the leaf space of H|_{V_{x}} is an embedding.

Existence of U_{x} is guaranteed by the fact that P_{k_{r}} − Σ_{f} is tangent to H. Then S(U_{x} ∩P_{k_{r}})∩V_{x} is an embedded submanifold (it is isomorphic to U_{x}^{2} ×ℝ). Now, since bU_{r} ∩ P_{k_{r}} ∩ L_{j} is compact, it is covered by a finite number of U_{x}’s, say by U_{x_{1}}, …, U_{x_{n}}. Moreover, we may assume that for each 1 ≤ ℓ ≤ n, there exists a relatively compact refinement V_{ℓ} of U_{x_{ℓ}}, such that ∪_{ℓ}V_{ℓ} also cover bU_{r} ∩P_{k_{r}} ∩L_{j}. Then the compact sets S(\overline{V_{ℓ}} ∩ P_{k_{r}}) ∩ L_{j}, ℓ = 1, …, n and U_{r} ∩ P_{k_{r}} cover N_{r}, and are contained in the submanifolds S(U_{x_{ℓ}} ∩ P_{k_{r}}) ∩ V_{x_{ℓ}}, ℓ = 1, …, n and P_{k_{r}} respectively. Because P_{k_{r}} is transverse to F, and because H is tangent to F, the submanifold S(U_{x_{ℓ}} ∩ P_{k_{r}}) ∩ V_{x_{ℓ}} is transverse to F as well (since U_{x}^{2} is transverse to F). Moreover, by construction, the submanifolds S(U_{x_{ℓ}} ∩ P_{k_{r}}) ∩ V_{x_{ℓ}} and P_{k_{r}} have codimension at least one.

It follows from Lemma 6.5 and Lemma 6.3 that the stable set of L_{j} ∩Σ_{f} with respect to (∇_{F}f)|_{L_{j}} is contained in a finite union of compact subsets of embedded submanifolds intersecting F transversely, and having their codimension at least equal to one. We can thus use Theorem 1.13 and Remark 1.14 to conclude.

End of the proof of Theorem 6.1.

7 Examples of open foliated manifolds

The very first class of examples of open foliated manifolds consists of the products (M, F) × ℝ. Let f be any positive, proper, F-generic function on (M, F) (such a function always exists). The function g : M ×ℝ → ℝ : (x, t) ↦ f(x)+t^{2} has no leafwise local maxima, and satisfies the hypotheses of Definition 0.3. As implied by the following theorem due to Palmeira, foliations of the type (M, F) × ℝ include an important class of plane foliations (i.e. foliations whose leaves are diffeomorphic to some Euclidean space).

Theorem 7.1 ([18]) If F is a transversely orientable plane foliation on an orientable n-dimensional manifold M (with n ≥ 3), with finitely generated fundamental group, such that all leaves are closed, then there exists a two dimensional surface S and a plane foliation F_{0} on S such that F is conjugate to the product of F_{0} with ℝ^{n−2}.

Another class of examples is described in the following lemma.

Lemma 7.2 Let π : M → B be a locally trivial fibration with compact fiber L and open base, and let F be a foliation on M that is transverse to the fibers of π, in the sense that T_{x}F + Ker π_{∗_{x}} = T_{x}M for all x in M. Then the foliated manifold (M, F) is open.

Proof. Let g : B → [0, ∞) be a proper Morse function without local maxima. It is sufficient to prove that if f : M → [0, ∞) is sufficiently close to π^{∗}g in the fine C^{∞} topology, then f is proper and has no leafwise local maxima. It is easy to see that a function that is C^{0}-close to a proper function is proper as well. Let {x^{j}; j ≥ 1} be the set of critical points of g. For every j, let x_{1}^{j}, …, x_{k}^{j} be local coordinates about x^{j}, defined on a neighborhood U^{j} of x^{j} in B, for which

g(x_{1}^{j}, …, x_{k}^{j}) = (x_{1}^{j})^{2} + g(0, x_{2}^{j}, …, x_{k}^{j}) .

Assume also that U^{j} is small enough to guarantee existence of a trivialization ϕ : π^{−1}(U^{j}) → U^{j} × L, with the property that the image of each local sections σ_{ℓ}^{j} : U^{j} → π^{−1}(U^{j}) : x ↦ ϕ^{−1}(x, ℓ), ℓ ∈ L is contained in a leaf of F. A function f whose 2-jet is sufficiently close to that of π^{∗}g satisfies the following properties.

- The leafwise critical points of f are all contained in π^{−1}(∪_{j}U^{j}).

- For all j ≥ 1, and for all ℓ in L, the second derivative of f ∘ σ_{ℓ}^{j} in the direction of x_{1}^{j} is strictly positive on U^{j}.

Then, if y is a leafwise critical point of f, the foliated second differential of f at y may not be negative definite, that is, y may not be a leafwise local maximum of f.

Remark 7.3 It is not know to us whether locally trivial fibrations with open fibers are always open as foliated manifolds. Nevertheless, such foliated manifolds satisfy the conclusion of Theorem 0.4, as implies by Theorem 0.1 (cf. [1, 2]).

We will now describe a general procedure that allows one to construct open foliated manifolds. We need to recall the notion of Novikov component of a codimension one foliation F on a closed manifold M.

Definition 7.4 ([17]) Two points x and y in M are said to be equivalent with respect to F if either F_{x} = F_{y}, or the foliation F admits a closed transversal that contains both x and y.

It is not difficult to verify that this defines an equivalence relation on M.

Definition 7.5 A Novikov component of the foliation F is an equivalence class for this equivalence relation.

Example 7.6 The Reeb foliation on S^{3} has three Novikov components. One of them is the torus leaf. The other two are two open solid tori bounded by the torus leaf.

Theorem 7.7 ([17]) A Novikov component is either a compact leaf or an open submanifold whose boundary is a finite union of compact leaves which are themselves Novikov components.

The following result is due to Ferry and Wasserman.

Theorem 7.8 ([6]) For a codimension one foliation F on a closed manifold M, the following statements are equivalent.

- F has one Novikov component.

- There exists an F-generic function f : M → ℝ with no leafwise degenerate critical points.

Consider now a codimension one foliation F with one Novikov component on a closed (n + 1)-dimensional manifold M. Let f : M → ℝ be an F-generic function without leafwise degenerate critical points. Let Γ ⊂ M be the set of leafwise local maxima of f. Notice that since f has no leafwise degenerate critical points, its singular locus (Definition 3.2) is necessarily transverse to F (cf. Section 3.1 or [6]). It is therefore a finite union of embedded circles intersecting F transversely; the set Γ is the union of some of them.

Lemma 7.9 The foliated manifold (M′ = M − Γ, F′ = F|_{M−Γ}) is open.

Proof. We already have a bounded below F-generic function f|_{M′} on M′ with no leafwise local maxima. The only thing that needs to be done is to modify f|_{M′} so as to make it proper. Take a tubular neighborhood U of Γ in M. The set U is the image of an embedding e : E → M, defined on the total space of a rank-n vector bundle p : E → Γ, such that e ∘ s = Id_{Γ}, where s : Γ → E is the zero section. Since Γ is transverse to F, we may assume, without loss of generality, that the foliation F|_{U} corresponds, under the map e, to the foliation of E by the fibers of p. Such a tubular neighborhood is called hereafter a foliated tubular neighborhood. Let g be a Riemannian metric on the bundle E. Let θ : ℝ → [0, 1] be a function with compact support whose only critical value, aside from 0, is a global maximum achieved at the point 0. Consider the function

f′ : M′ → ℝ : x ↦ \{ \begin{matrix}f(x) + \frac{θ(g(e^{−1}x, e^{−1}x))}{g(e^{−1}x, e^{−1}x)} & for x in U − Γ \\ f(x) & otherwise . \end{matrix}

The function f′ is proper and bounded below. Moreover, provided U is small enough, its leafwise critical points are exactly those of f|_{M′}. Thus f′ has no leafwise local maxima.

Remark 7.10 Making f|_{M−Γ} proper requires the set Γ of leafwise local maxima of f to be a union of circles. If on the contrary, Γ has a line segment as one of its connected components, as might be the case if the function f had leafwise degenerate critical points, it would not be possible to make f|_{M−\overline{Γ}} proper without creating leafwise local maxima.

Remark 7.11 It is a classical result due to Rummler and Sullivan that a transversely orientable, codimension one foliation has one Novikov component if and only if it is geometrically taut, that is, if and only if M admits a Riemannian metric with respect to which all the leaves of F are minimal submanifolds (a proof of this result can be found in [4]).

As was suggested to us by Takashi Tsuboi, interesting foliations with one Novikov component are obtained by forming connected sums along closed transversals. Let M_{1} and M_{2} be (n + 1)-dimensional manifolds endowed with codimension one foliations F_{1} and F_{2} respectively. For i = 1, 2, let c_{i} : S^{1} → M_{i} be an embedding transverse to F_{i}. Suppose that the normal bundle of c_{i}(S^{1}) is trivial (which it is when the foliated manifold (M_{i}, F_{i}) is orientable). Consider a foliated tubular neighborhood U_{i} of c_{i}(S^{1}). The set U_{i} is the image of an embedding e_{i} : S^{1} × ℝ^{n} → M_{i} that coincides with c_{i} on S^{1} × {0}, and that maps the “vertical” foliation of S^{1} × ℝ^{n} isomorphically onto the foliation F_{i}|_{U_{i}}. The connected sum of the foliated manifolds (M_{1}, F_{1}) and (M_{2}, F_{2}) along the closed transversals c_{1} and c_{2}, denoted by (M_{1}, F_{1})_{c_{1}}#_{c_{2}}(M_{2}, F_{2}), is defined to be the foliated manifold obtained by gluing M_{1} − c_{1}(S^{1}) to M_{2} − c_{2}(S^{1}) along U_{1} − c_{1}(S^{1}) and U_{2} − c_{2}(S^{1}) via the isomorphism

U_{1} − c_{1}(S^{1}) → U_{2} − c_{2}(S^{1}) : e_{1}(t, x) ↦ e_{2}(t, \frac{x}{∥x∥^{2}} ) .

If both F_{1} and F_{2} have one Novikov component, then their connected sum has one Novikov component as well. Notice that one could as well perform the connected sum along disjoint unions of embedded circles, c_{1}^{1} ∪ … ∪ c_{1}^{k} in M_{1}, and c_{2}^{1} ∪ … ∪ c_{2}^{k} in M_{2}.

Consider, for instance, a manifold of the type M = S^{1} × F, where F is some n-dimensional manifold. Let π denote the natural projection M → S^{1}, and let F denotes the (trivial) foliation of M by the fibers of π. Let also c_{1}, c′_{1}, c_{2} : S^{1} → M be disjoint embeddings such that π ∘ c_{1} = π ∘ c′_{1} = Id_{S^{1}}, and such that π ∘ c_{2} coincides with the double cover e^{it} ↦ e^{i2t} (alternatively, one could suppose that π ∘ c_{1} = π ∘ c′_{1} and π ∘ c_{2} are different covers of S^{1} of the type e^{it} ↦ e^{int}). Then the connected sum of (M, F) with itself along the closed transversals c_{1} ∪ c′_{1} and c_{2} ∪ c′_{1} is a foliated manifold with one Novikov component, and with dense leaves. Indeed, a leaf of the connected sum corresponds to an equivalence class for the equivalence relation on S^{1} generated by e^{it} ∼ e^{i2t}. The open foliated manifold obtained by removing from M the set of closed curves along which some given F-generic function without any leafwise degenerate critical point achieves leafwise local maxima has dense leaves as well. Besides, if F is an open manifold, it is not necessary to remove something to achieve openness.

Lemma 7.12 If the manifold F is open, then the foliated manifold (M, F)_{c_{1}∪c′_{1}} #_{c_{2}∪c′_{1}}(M, F) is open.

Proof. Let f : F → [0, ∞) be a proper Morse function without local maxima. Let a, b be distinct noncritical values of f. Let γ : S^{1} → F be an injective map whose image is contained in the level f^{−1}(a). Let q (respectively q′) be a point in f^{−1}(a) − γ(S^{1}) (respectively f^{−1}(b)). Suppose that the closed transversal c_{1} (respectively c′_{1}) coincides with the map S^{1} → M : t ↦ (t, q) (respectively S^{1} → M : t ↦ (t, q′)), and that c_{2} coincides with the map S^{1} → M : t ↦ (2t, γ(t)). Suppose also that on the tubular neighborhoods U_{1}, U′_{1} and U_{2} of c_{1}, c′_{1} and c_{2} respectively, along which the connected sum is performed, the function p^{∗}f corresponds to some function

S^{1} × ℝ^{n} → ℝ : (t, x_{1}, …, x_{n}) ↦ c + h(x_{n}) ,

where p denotes the projection M → F, where h : ℝ → ℝ is an embedding, and where c is either a or b. The functions f|_{M−(c_{1}(S^{1})∪c′_{1}(S^{1}))} and f|_{M−(c_{2}(S^{1})∪c′_{1}(S^{1}))} assemble into a proper function g : M_{c_{1}∪c′_{1}}#_{c_{2}∪c′_{1}}M → [0, ∞) that coincides with f on (M − (U_{1} ∪ U′_{1})) ∐(M − (U_{2} ∪ U′_{1})), and whose critical locus in U_{1} − (c_{1}(S^{1}) ∪ c′_{1}(S^{1})) ≃ U_{2} − (c_{2}(S^{1}) ∪ c′_{1}(S^{1})) is made of two disjoint transverse closed curves, consisting of leafwise critical points of leafwise index m − 1 and 1 respectively. Figure 1 shows how to construct g on the connected sum of a fiber in the tubular neighborhood U_{1} (or U′_{1}) with the corresponding fiber in the tubular neighborhood U_{2} (or U′_{1}).

ℝ^{index m − 1}^{index 1}  ^{a}

Figure 1: The function g on the connected sum of two corresponding fibers

8 Application to Poisson geometry

Let us recall that a regular Poisson manifold can be described as a foliated space (M, F) endowed with a leafwise symplectic structure, that is, a section of the second exterior power of the cotangent bundle T^{∗}F of the foliation F, whose restriction to each leaf of F is a symplectic form (cf. [5]). The question of existence of such a structure on a given foliated manifold as been approached in a previous paper [2] (see also [1]), where examples of foliations are presented that do not support any leafwise symplectic structure although the obvious obstructions vanish. On the other hand, as explained below, a leafwise symplectic structure on the foliated manifold (M, F) is a solution of a certain open, foliated invariant differential relation.

Consider a foliated manifold (M, F). A section of the bundle Λ^{k}T^{∗}F is called a tangential differential k-form. A tangential differential 2-form α is said to be nondegenerate if for every point x in M, the skewsymmetric bilinear form α(x) on T_{x}F is nondegenerate. The usual exterior differential restricts naturally into a map d_{F} : Γ(Λ^{⋆}T^{∗}F) → Γ(Λ^{⋆+1}T^{∗}F) (cf. [11]). The corresponding cohomology

H^{⋆}(F) = \frac{\{d_{F} − closed ⋆ −forms \}}{\{d_{F} − exact ⋆ −forms \}} ,

is called the tangential de Rham cohomology of the foliated manifold (M, F). There is a natural affine fibration L : (T^{∗}F)^{1} → Λ^{2}T^{∗}F defined by L(j^{1}α(x)) = d_{F}α(x). Choose a d_{F}-closed tangential differential 2-form θ. Define Ω_{θ} to be the differential relation Ω_{θ} = { j^{1}α(x) ; θ(x) + L(j^{1}α(x)) is nondegenerate }. A solution of Ω_{θ} is thus a tangential differential 1-form α such that d_{F}α + θ is a leafwise symplectic structure belonging to the same tangential de Rham cohomology class as θ. Thus, the set of solutions, modulo the set of d_{F}-closed 1-forms, parameterizes the set of leafwise symplectic structures lying in the class [θ]. On the other hand, since the map L is an affine fibration, the space of sections of Ω_{θ} is weakly homotopy equivalent to the space of leafwise nondegenerate 2-forms.

Proposition 8.1 ([7]) The relation Ω_{θ} is open and invariant under foliated isotopies.

Proof. That Ω_{θ} is open follows directly from its definition. The second assertion relies on considering the “right” lift for isotopies, that is :

φ_{t} · α = φ_{t}^{∗}α + ∫_{0}^{t} φ_{s}^{∗} (i(X_{s})θ) ds ,

where α is an element of T^{∗}F, and where X_{t} denotes the time-dependent vector field associated to the isotopy φ_{t}. If α is a tangential differential 1-form for which the tangential differential 2-form d_{F}α + θ is leafwise nondegenerate, then φ_{t}^{∗}(d_{F}α+θ) is leafwise nondegenerate as well (while the form φ_{t}^{∗}(d_{F}α)+θ might very well be leafwise degenerate), and

φ_{t}^{∗}(d_{F}α + θ) = d_{F}φ_{t}^{∗}α + θ + (φ_{t}^{∗}θ − θ)
= d_{F}φ_{t}^{∗}α + θ + ∫_{0}^{t} \frac{d}{ds}φ_{s}^{∗}θ|_{s} ds
= d_{F}φ_{t}^{∗}α + θ + ∫_{0}^{t} φ_{s}^{∗}L_{X_{s}}θ ds
= d_{F}φ_{t}^{∗}α + θ + ∫_{0}^{t} φ_{s}^{∗}d_{F}i(X_{s})θ ds
= d_{F} (φ_{t}^{∗}α + ∫_{0}^{t} φ_{s}^{∗}i(X_{s})θ ds) + θ .

Thus, Theorem 0.4 applies to the relation Ω_{θ}.

Theorem 8.2 Let (M, F) be an open foliated manifold. Given a d_{F}-closed tangential differential 2-form θ, any family β_{s}, s ∈ [0, 1]^{p} of leafwise nondegenerate 2-forms is homotopic to a family of leafwise symplectic structures cohomologous to θ. Moreover, if β_{s} is already symplectic and cohomologous to θ for s in ∂([0, 1]^{p}), the homotopy may be chosen to be stationary for those parameters. Equivalently, the inclusion of the space of leafwise symplectic structures cohomologous to θ into the space of leafwise nondegenerate 2-forms is a weak homotopy equivalence

Theorem 8.2 implies the following existence and uniqueness result for leafwise symplectic structures.

Corollary 8.3 On an open foliated manifold any leafwise nondegenerate 2-form can be deformed into a leafwise symplectic form (with prescribed tangential de Rham cohomology class). Moreover, if two cohomologous leafwise symplectic structures can be joined by a path of leafwise nondegenerate 2-forms, they can also be joined by a path of cohomologous leafwise symplectic forms.

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