A generalisation of the Morse inequalities

Mohan Bhupal

1 Introduction

In [1] Arnold calls a Legendrian embedding f: L → J^{1}M, into the 1-jet bundle, that can be obtained from the zero section by a smooth deformation through Legendrian embeddings a quasi-function. Intersections of f with M_{0} ×ℝ, where M_{0} denotes the zero section of T^{∗}M, are called critical points of f and these are said to be nondegenerate if the intersection is transverse. It can be shown that every such quasi-function is of the form f = ψ|_{M_{0}×{0}}, where ψ = ψ^{1} for some contact isotopy (ψ^{t})_{0≤t≤1} of J^{1}M. Under the assumption that M is closed (that is, compact and boundaryless) and a Riemannian metric on M has been chosen, we show that to each nondegenerate critical point c ∈ L of f we may naturally assign a number

μ(c, ψ^{t}) ∈ \frac{n}{2} + ℤ

as the Maslov index of the path γ: t ↦ ψ^{t}((ψ^{1})^{−1}f(c)). Assuming that all critical points are nondegenerate we denote by p_{k} the number of critical points c ∈ L such that μ(c, ψ^{t}) = k and by b_{k} the k th Betti number of M (with coefficients in any field). We prove the following:

Theorem 1.1 Let f: L → J^{1}M be a quasi-function on a closed Riemannian manifold M of dimension n having only nondegenerate critical points. Then

p_{\frac{n}{2} −k} − p_{\frac{n}{2} −k+1} + ⋯ ≥ b_{k} − b_{k−1} + ⋯ + (−1)^{k}b_{0}

for every k ∈ ℤ.

Theorem 1.1 is a refinement of a theorem by Chekanov announced by Arnold in [1] (see also Chekanov [5]). This bounds the number of critical points, in the nondegenerate case, below by the sum of the Betti numbers. A proof of Chekanov’s Theorem is also given by Chaperon [4] and is based on a result by Théret [16] concerning the existence of a generating function. Our proof is independent of their work.

A method of generating contact isotopies is by using a time-dependent Hamiltonian H: [0, 1]×J^{1}M → ℝ. Indeed all contact isotopies may be obtained in this way. A special instance is given by the following: given a Morse function f: M → ℝ, let H(x, y, z) = −f(x) define a time-independent Hamiltonian. The critical points of f are in one-to-one correspondence with the critical points of the associated quasi-function. In particular the Morse index ind_{H}(x_{0}) of a critical point x_{0} ∈ M of f is related to the Maslov index by

μ(x_{0}, ψ_{H}^{t}) = \frac{n}{2} − ind_{H}(x_{0})

and thus in this special case Theorem 1.1 reduces to the classical Morse inequalities.

Our methods also refine a related theorem in the symplectic setting proved by Hofer [9] and Laudenbach and Sikorav [11]. This pertains to finding a lower bound for the number of points of intersection of the zero section of the cotangent bundle with its image under a time-dependent Hamiltonian symplectomorphism. An outline of our approach follows.

Let ψ^{t} be a contact isotopy of J^{1}M with Hamiltonian H: [0, 1] × J^{1}M → ℝ. When M = ℝ^{n} we may associate to H an action functional as follows. Consider the path space

P = {c = (x, y): [0, 1]→T^{∗}ℝ^{n} | y(0) = 0} ,

and, given c ∈ P, let z = z_{c}: [0, 1] → ℝ be the unique solution of

ż = ⟨y, ẋ⟩ − H(t, x, y, z) z(0) = 0

(assuming such a solution exists). Define the action functional Φ_{H}: P → ℝ by

Φ_{H}(c) = ∫_{0}^{1} (⟨y, ẋ⟩ − H(t, x, y, z)) dt.

A calculation shows that the fibre critical points of Φ_{H} with respect to the fibration P → ℝ^{n}: (x, y) ↦ x(1) generate the Legendrian submanifold ψ^{1}(L_{0}).

We consider discretisations of the form Φ_{H}^{N}: P^{N} → ℝ for N ∈ ℕ which also have this property. Here

P^{N} = {c^{N} = (x_{0}, …, x_{N}, y_{1}, …, y_{N}) ∈ ℝ^{(2N+1)n} | x_{i}, y_{j} ∈ ℝ^{n}} ,

which is fibred over ℝ^{n} by P^{N} → ℝ^{n}: c^{N} ↦ x_{N}, can be thought of as a space of discrete paths in ℝ^{2k}, and Φ_{H}^{N} is the discrete action functional which is given by

Φ_{H}^{N}(c^{N}) = ∑_{j=1}^{N} (⟨y_{j}, x_{j} − x_{j−1}⟩ − V_{j−1}(x_{j−1}, y_{j}, z_{j−1})).

The functions V_{j}: ℝ^{2n+1} → ℝ are constructed by using the contact isotopy ψ^{t}, and the z_{j} are given by the iterative rule

z_{0} = 0, z_{j} = ⟨y_{j}, x_{j} − x_{j−1}⟩ − V_{j−1}(x_{j−1}, y_{j}, z_{j−1}) + z_{j−1}, j > 0.

We show that such discretisations always exist for N sufficiently large and H having compact support.

A path c ∈ P is a critical point of Φ_{H} if and only if (c, z_{c}) is a trajectory of ψ^{t} starting in L_{0} × {0}, where L_{0} denotes the zero section of T^{∗}ℝ^{n}, and ending in L_{0} × ℝ. Similarly, a discrete path c^{N} ∈ P^{N} is a critical point of Φ^{N} if and only if (c^{N}, {z_{j}}) corresponds to a “broken trajectory” of ψ^{t} with (x_{0}, y_{0}, z_{0}) ∈ L_{0} × {0} and (x_{N}, y_{N}, z_{N}) ∈ L_{0} × ℝ. Furthermore given such a path c^{N}, critical for Φ^{N}, and such that ψ^{1}(L_{0} × {0}) intersects L_{0} × ℝ transversally at (x_{N}, y_{N}, z_{N}), d^{2}Φ_{H}^{N}(c^{N}) will be nondegenerate. We show that, given a critical point c ∈ P with the same property, for sufficiently large N the signature of the second variation of the discretisation is given by:

Proposition 1.2

sign d^{2}Φ_{H}^{N}(c^{N}) = 2μ(c^{N}, ψ^{t}).

This generalises a theorem of Robbin and Salamon in [13] for quadratic Hamiltonians. Also, it strengthens the theorem of Viterbo [17] which states that, in the symplectic case, the difference of the signature of the second variation of Φ^{N} at two critical points is independent of N, but at the cost of having to take N sufficiently large.

The action functional and its discretisation are extended to the 1-jet bundle of a closed manifold M as follows. First, M is embedded in ℝ^{k} for a suitable k. Next, a lift H̃: J^{1}ℝ^{k} → ℝ of H is chosen such that H̃|_{J^{1}ℝ^{k}|_{M}} = H ∘ ρ where ρ: J^{1}ℝ^{k}|_{M} → J^{1}M is the natural projection map. Finally, the appropriate action functional is defined to be ΦH̃ restricted to the space of paths having x(1) ∈ M, and similarly for the discretisation. Theorem 1.1 follows from generalising Proposition 1.2 and appealing to stabilised Morse theory (the Conley index).

The author would like to thank his doctoral supervisor, D. Salamon, for his help.

2 Contact structures

In this section we recall some basic notions of contact structures. A contact structure on a 2n + 1-dimensional manifold Q is a maximally nonintegrable field of hyperplanes ξ ⊂ T Q. If we assume that ξ is transversally orientable, that is ξ is given by the kernel of some 1-form α, then the contact condition can be stated as

α ∧ (dα)^{n} ≠ 0.  (1)

In this case α is said to be a contact form for the contact structure ξ. An important equivalent formulation of the contact condition (1) is that dα is a nondegenerate 2-form on ξ.

Given a contact manifold (Q, ξ) and an integral submanifold L ⊂ Q note that, for each p ∈ Q, the tangent space T_{p}L is an isotropic subspace of the symplectic vector space (ξ_{p}, dα_{p}), that is, dα_{p} vanishes on T_{p}L. This implies, in particular, that the dimension of L can be at most n. In such a case L is said to be Legendrian.

A diffeomorphism ψ: Q → Q which preserves the oriented field of hyperplanes ξ is called a contactomorphism, that is, equivalently, if

ψ^{∗}α = e^{g}α

for some function g: Q → ℝ.

Note that if ψ: Q → Q is a contactomorphism, then, by a direct calculation, its derivative restricts to give a linear conformal symplectomorphism

dψ(p)|_{ξ_{p}}: (ξ_{p}, dα_{p}) → (ξ_{ψ(p)}, dα_{ψ(p)}),

for every p ∈ Q.

A smooth family (ψ^{t})_{0≤t≤1} of diffeomorphisms ψ^{t}: Q → Q with ψ^{0} = id is called an isotopy of Q. When each ψ^{t} is a contactomorphism the family ψ^{t} is called a contact isotopy.

A contact vector field is defined to be a vector field X: Q → T Q which satisfies

L_{X}α = hα

for some function h: Q → ℝ.

The contact Hamiltonian associated to a contact vector field X: Q → T Q is defined to be the function H: M → ℝ given by

H = −ι(X)α.  (2)

It is easy to check that every function H: Q → ℝ is the contact Hamiltonian of a unique contact vector field X = X_{H}: Q → T Q.

1-jet bundles

The standard contact structure on the 1-jet bundle J^{1}M = T^{∗}M ×ℝ of an n-dimensional manifold M is given as kernel of the 1-form

α = dz − λ_{can}.

Here z denotes the standard coordinate on ℝ and λ_{can} the canonical 1-form on T^{∗}M which is given by the defining property σ^{∗}λ_{can} = σ for every 1-form σ: M → T^{∗}M.

Let ψ^{t} be a contact isotopy of J^{1}M and X: [0, 1] × J^{1}M → T J^{1}M be the associated time-dependent contact vector field defined by

\frac{d}{dt}ψ^{t}(p) = X(t, ψ^{t}(p)).

Associated to X is the time-dependent Hamiltonian H: [0, 1] × J^{1}M → ℝ given by (2). We express this in local coordinates.

About any point (c, z) ∈ J^{1}M one can construct a distinguished set of local coordinates x_{1}, …, x_{n}, y_{1}, …, y_{n}, z in which α takes the form

α = dz − ∑_{i} y_{i} dx_{i}.

In these coordinates it is easy to check that X is given by Hamilton’s equations:

ẋ_{i} = \frac{∂H}{∂y_{i}}  (3a)
ẏ_{i} = −\frac{∂H}{∂x_{i}} − y_{i} \frac{∂H}{∂z}  (3b)
ż = ⟨y, ẋ⟩ − H  (3c)

and thus ψ^{t} is the time-t map of the flow associated to Hamilton’s equations for H.

3 The variational principle

In this section we describe an abstract method for generating all solutions of Hamilton’s equations subject to c(0) ∈ M_{0}, the zero section of T^{∗}M, and z(0) = 0. Let H: [0, 1] × J^{1}M → ℝ be a time-dependent Hamiltonian. To ensure that (4) below has a solution z: [0, 1] → ℝ for every smooth path c: [0, 1] → T^{∗}M assume that ∂_{z}H is bounded. Let π: P_{M} → M be the fibre bundle where

P_{M} = {c: [0, 1]→T^{∗}M | c(0) ∈ M_{0}}

is a space of smooth paths and π is the projection c ↦ π_{T^{∗}M}c(1) ∈ M. To each path c ∈ P_{M} associate a function z = z_{c}: [0, 1] → ℝ given as the unique solution of

ż(t) = λ_{can}(ċ(t)) − H(t, c(t), z(t)),  z(0) = 0.  (4)

Now define the action functional Φ_{H}: P_{M} → ℝ by

Φ_{H}(c) = z(1) = ∫_{0}^{1} (λ_{can}(ċ) − H(t, c, z)) dt.

A point c ∈ P_{M} is called a fiber critical point of Φ_{H} if the differential of Φ_{H} disappears along T_{c}(P_{M})_{π(c)}, the tangent space at c of the fibre over π(c).

Proposition 3.1 There is a one-to-one correspondence between the set of fibre critical points of Φ_{H} and the solutions of Hamilton’s equations subject to c(0) ∈ M_{0}, z(0) = 0. Furthermore, c ∈ P_{M} is fibre critical if and only if

ι(ċ)dλ_{can} + dH_{t,z}(c) + ∂_{z}H(t, c, z)λ_{can} = 0  (5)

Proof. Let (−ε, ε) → P_{M}: s ↦ c_{s} be a parametrised family of paths and abbreviate c_{0} = c. Denote by γ: [0, 1] → c^{∗}T (T^{∗}M) the associated variational field:

γ(t) = \frac{∂}{∂s}|_{0} c_{s}(t).

Define z_{s}: [0, 1] → ℝ for each s by (4) and denote the derivative with respect to s at s = 0 by ζ. Now ζ evaluated at t ∈ [0, 1] is given by the expression

ζ(t) = \frac{∂}{∂s}|_{0} ∫_{0}^{t} c_{s}^{∗}(λ_{can} − H_{z_{s}} dτ)
= ∫_{0}^{t} c^{∗}L_{γ}(λ_{can} − H_{z} dτ) − ∂_{z}Hζ dτ
= ∫_{0}^{t} (ι(γ)(−ι(ċ)dλ_{can} − dH_{τ,z}) − ∂_{z}Hζ) dτ + λ_{can}(γ(t))

where the third equality follows from Cartan’s identity for the Lie derivative. Thus the 1-form α applied to the vector field (γ, ζ) satisfies

α(γ(t), ζ(t)) = ∫_{0}^{t} (ι(γ)(−ι(ċ)dλ_{can} − dH_{τ,z} − ∂_{z}Hλ_{can}) − ∂_{z}Hα(γ, ζ)) dτ
α(γ(0), ζ(0)) = 0.

This we can solve explicitly. Evaluating at t = 1 we find

α(γ(1), ζ(1)) = ∫_{0}^{1} e^{∫_{t}^{1} −∂_{z}H dτ} ι(γ)(−ι(ċ)dλ_{can} − dH_{t,z} − ∂_{z}Hλ_{can}) dt.

Thus it follows that c ∈ P_{M} is fibre critical if and only if (5) holds which, in local coordinates, is just equations (3a, b). □

Note that at a fibre critical point c the differential of Φ_{H} is given by

dΦ_{H}(c)(γ) = λ_{can}(γ(1)).

It thus follows that c is a critical point only if c(1) ∈ M_{0}.

4 Variational families

The notion of a variational family formalises the variational principle that was considered in the previous section. In this section we consider some basic properties of variational families in general. We start by briefly recalling some ideas from the theory of symplectic reduction.

Suppose that (P, ω) is a symplectic manifold and that W ⊂ P is a coisotropic submanifold. This means that T_{p}W^{ω} is contained in T_{p}W for each p ∈ W where T_{p}W^{ω} denotes the symplectic complement of T_{p}W. Then the subspaces T_{p}W^{ω} determine an isotropic distribution T W^{ω} in T P which by the closedness of ω is integrable. It follows from Frobenius’ theorem that W is foliated by isotropic leaves. If we assume that the quotient space P′ = W/∼ is a manifold, where the equivalence relation is given by: p_{0} ∼ p_{1} if p_{0} and p_{1} lie in the same leaf, then we note that the P′ is naturally a symplectic manifold.

We next recall that if L is a Lagrangian submanifold of P which intersects W cleanly, that is, L ∩ W is a submanifold of P and T_{p}(L ∩ W) = T_{p}L ∩ T_{p}W for all p ∈ L ∩ W, then the image of L in P′ is also Lagrangian, but may only be immersed.

To pass from Lagrangian immersions to Legendrian immersions we use the process of contactisation. This applies to all symplectic manifolds (P, ω) where the symplectic form is given as the differential of a 1-form −λ. Such manifolds are said to be exact. A contactisation of an exact symplectic manifold (P, ω) is a contact manifold (P ×ℝ, dz−λ) where λ is a 1-form satisfying the relation ω = −dλ. Here z denotes the coordinate on ℝ. Given an exact symplectic manifold (P, ω), a contactisation of a Lagrangian immersion f: L → P is a Legendrian immersion f̃: L → P × ℝ lifting f. If the Lagrangian immersion f: L → P is exact, that is, f^{∗}λ = dS for some function S: L → ℝ, then the lift L → P × ℝ: c ↦ (f(c), S(c)) is Legendrian. Conversely it is easy to see that if f admits a lift then it must be exact. Note however that the contactisation of an exact Lagrangian immersion is not unique since the function S is unique only up to the addition of a locally constant function.

We now give the definition of a variational family. This is a pair consisting of a fibre bundle π: E → X and a smooth function Φ: E → ℝ. The variational is called transversal if the graph of dΦ in T^{∗}E intersects the conormal bundle of the fibres

N_{E} = {(c, η) ∈ T^{∗}E | η ∈ (ker dπ(c))^{⊥}}

transversally. For a transversal variational family (E, Φ) the set of fibre critical points is given by

C_{E,Φ} = {c ∈ E | dΦ(c) ∈ (ker dπ(c))^{⊥}}

which by transversality is a manifold. To each fibre critical point we can uniquely associate a covector v^{∗} ∈ T_{π(c)}^{∗}X by the relation v^{∗} ∘ dπ(c) = dΦ(c). This gives rise to a map ι_{Φ} : C_{E,Φ}→J^{1}X given by mapping c ∈ C_{E,Φ} to the 1-jet (π(c), v^{∗}, Φ(c)).

The following proposition is due to Hörmander [10]. We include here a proof for the sake of completeness.

Proposition 4.1 Given a transversal variational family (E, Φ), the map ι_{Φ} : C_{E,Φ} → J^{1}M is a Legendrian immersion.

Proof. Note that the graph of dΦ is a Lagrangian submanifold of T^{∗}E and N_{E} is a coisotropic submanifold. Note also that the quotient N_{E}/∼ can be identified with T^{∗}X. It follows from the above discussion that the map ι′_{Φ} : C_{E,Φ} → T^{∗}X, given by composing ι_{Φ} with the projection J^{1}X → T^{∗}X, is a Lagrangian immersion. Moreover the pullback of λ_{can} by ι′_{Φ} is given by

ι′_{Φ}^{∗}λ_{can} = v^{∗} ∘ d(π|_{C_{E,Φ}}) = d(Φ|_{C_{E,Φ}})

and so ι′_{Φ} is an exact immersion. Contactising, it follows that ι_{Φ} is a Legendrian immersion. □

Denote by L_{Φ} the image of ι_{Φ} and by X_{0} the zero section of T^{∗}X. Clearly, a point c ∈ C_{E,Φ} is a critical point of Φ if and only if ι_{Φ}(c) is in L_{Φ} ∩ X_{0} × ℝ. Moreover, when E is a finite dimensional vector bundle we have:

Lemma 4.2 A point c ∈ crit Φ is a nondegenerate critical point of Φ if and only if L_{Φ} and X_{0} × ℝ intersect transversally at ι_{Φ}(c).

The proof of this uses the following elementary fact:

Lemma 4.3 Let (V, ω) be a symplectic vector space and N ⊂ V a coisotropic subspace. Then given two Lagrangian subspaces Λ_{0} and Λ_{1} satisfying

Λ_{0} ⊂ N, Λ_{1} ∩ N^{ω} = {0},

Λ_{0} is transverse to Λ_{1} if and only if, in the quotient, the reduced spaces \overline{Λ}_{0} and \overline{Λ}_{1} are transversal.

Proof of Lemma 4.2. Apply Lemma 4.3 in case V = T_{p}(T^{∗}E), N = T_{p}N_{E}, Λ_{0} = T_{p}E_{0} and Λ_{1} = T_{p}(graph(dΦ)) where p = (c, dΦ(c)) and E_{0} denotes the zero section of T^{∗}E. □

Example 4.4 (Action) Let H: [0, 1] × J^{1}M → ℝ be a time-dependent Hamiltonian and Φ_{H}: P_{M} → ℝ the action functional given in Section 3. By Proposition 3.1 the pair (P_{M}, Φ_{H}) is a variational family for the Legendrian submanifold ψ^{1}(M_{0} × {0}), where ψ^{t} = ψ_{H}^{t} denotes the contact isotopy generated by H. Note that we do not say anything about transversality.

Restricted variational families

Let M be a submanifold of X and let L → J^{1}X: c ↦ (f(c), S(c)) be a Legendrian immersion transversal to J^{1}X|_{M}. We extend the notion of symplectic reduction to such Legendrian immersions as follows.

Evidently f: L → T^{∗}X is an exact Lagrangian immersion. Define f′: L′ → T^{∗}M, where L′ = f^{−1}(T^{∗}X|_{M}), by f′ = r ∘ f ∘ i. Here i: L′ ↪ L is the inclusion and r: T^{∗}X|_{M} → T^{∗}M is the natural projection. By transversality L′ is a manifold of the same dimension as M. Using r^{∗}λ_{can} = j^{∗}λ_{can} where j: T^{∗}X|_{M} ↪ T^{∗}X is the inclusion, it is now easy to see that f′ is again an exact Lagrangian immersion with f′^{∗}λ_{can} = d(S ∘ i). The lift L′ → J^{1}M: c ↦ (f′(c), S ∘ i(c)) we refer to as the reduction of (f, S).

Suppose that (π: E → X, Φ: E → ℝ) is a transversal variational family and denote by

(π_{M} = π|_{E_{M}} : E_{M}→M, Φ_{M} = Φ|_{E_{M}} : E_{M}→ℝ),

where E_{M} = E|_{M}, the variational family obtained by restricting to M. In the case of a finite dimensional vector bundle these are related by:

Lemma 4.5 Given a transversal variational family (E, Φ), the restricted variational family (E_{M}, Φ_{M}) is transversal if and only if L_{Φ} is transverse to J^{1}X|_{M}. Moreover, if both variational families are transversal then L_{Φ_{M}} is the reduction of L_{Φ}.

Proof. The first part of the lemma follows easily by choosing local coordinates on X in which M is linear and the second part follows by a direct calculation. □

Corollary 4.6 If both (E, Φ) and (E_{M}, Φ_{M}) are transversal then c ∈ E_{M} is a nondegenerate critical point of Φ_{M} if and only if L_{Φ} and T M^{⊥} × ℝ, where T M^{⊥} ⊂ T^{∗}X denotes the conormal bundle of M, intersect transversally at ι_{Φ}(c).

Proof. This follows immediately from Lemmas 4.2, 4.3 and 4.5. □

5 Continuous-time theory

In this section we assume that M is embedded in ℝ^{k} and that ψ^{t}: J^{1}M → J^{1}M is a given contact isotopy. We construct a variational family (P_{ℝ^{k}}, Φ_{H}) on ℝ^{k} whose restriction to M, (P, Φ), generates the Legendrian submanifold ψ^{1}(M_{0} × {0}) ⊂ J^{1}M, and consider the second variation of the function Φ.

5.1 Construction of variational family

Suppose that H: [0, 1] × J^{1}ℝ^{k} → ℝ is a time-dependent Hamiltonian with the property that for every x ∈ M

H(t, x, y, z) = H(t, x, y′, z) whenever y − y′ ⊥ T_{x}M.

Denoting by ρ: J^{1}ℝ^{k}|_{M}→J^{1}M the natural projection (x, y, z) ↦ (x, y|_{T_{x}M}, z), it follows that H descends to a time-dependent Hamiltonian H: [0, 1] × J^{1}M → ℝ given by

H|_{J^{1}ℝ^{k}|_{M}} = H ∘ ρ.  (6)

We insist that H_{t} is C^{1}-bounded for every t ∈ [0, 1]. This ensures that the Hamiltonian flows of H and H define global diffeomorphisms for all t ∈ [0, 1].

The next lemma shows how the Hamiltonian flow of H is related to that of H (cf. [2]).

Lemma 5.1 Let H be a time-dependent Hamiltonian on J^{1}ℝ^{k} and denote by φ^{t} the associated contact isotopy. Then the following are equivalent:

(i) H satisfies (6) for some function H: [0, 1] × J^{1}M → ℝ,

(ii) φ^{t} leaves J^{1}ℝ^{k}|_{M} invariant,

(iii) φ^{t} descends to a contact isotopy on J^{1}M.

Proof. (i) ⇒ (ii). Let X = (ẋ, ẏ, ż) denote the vector field generated by H. It is sufficient to show that X_{t}(p) ∈ T_{p}(J^{1}ℝ^{k}|_{M}) for all p ∈ J^{1}ℝ^{k}|_{M}. To see this note that, for any p = (x, y, z) ∈ J^{1}ℝ^{k}|_{M},

ẋ_{t}(p) = ∂_{y}H_{t}(x, y, z)
= ∂_{y}H_{t}(x, Π(x)y, z)
= Π(x)∂_{y}H_{t}(x, y, z).

Here T^{∗}M has been identified with T M, using the induced inner product, and Π(x) ∈ ℝ^{k×k} denotes the orthogonal projection onto T_{x}M. It follows ẋ_{t}(p) ∈ T_{x}M and hence X_{t}(p) ∈ T_{p}(J^{1}ℝ^{k}|_{M}) as required.

(ii) ⇒ (iii). We first claim that if a contactomorphism φ preserves J^{1}ℝ^{k}|_{M} then it has the property that

ρ(φ(p)) = ρ(φ(q)) whenever ρ(p) = ρ(q).  (7)

To see this note that the subspace N_{p} = T_{p}(J^{1}ℝ^{k}|_{M}) ∩ ξ_{p} is a coisotropic subspace of the symplectic vector space (ξ_{p}, dα_{p}), for each p ∈ J^{1}ℝ^{k}|_{M}, and that the symplectic complement is given simply by N_{p}^{ω} = T_{p}(ρ^{−1}(ρ(p))). Also note that dφ(p) restricts to a map from N_{p} to N_{φ(p)}. Property (7) now follows from the fact that the fibers of ρ are connected.

Consequently, we may define ψ^{t} : J^{1}M → J^{1}M by ρ ∘ φ^{t} ∘ j = ψ^{t} ∘ ρ, where j denotes the inclusion J^{1}ℝ^{k}|_{M} ↪ J^{1}ℝ^{k}. Clearly ψ^{t} is a diffeomorphism. That ψ^{t} is a contactomorphism follows from a direct calculation using the identity j^{∗}α = ρ^{∗}α.

(iii) ⇒ (i). Since φ^{t} descends to a contact isotopy ψ^{t} on J^{1}M it is clear that φ^{t} preserves J^{1}ℝ^{k}|_{M}. Also it follows that the contact vector fields X, X generated by the isotopies φ^{t}, ψ^{t} respectively are related by ρ_{∗}X = X. These satisfy ι(X)α = −H, ι(X)α = −H, where H is the Hamiltonian function generating ψ^{t}. A direct calculation now shows that H and H are related by (6). This completes the proof of the lemma. □

Remark 5.2 It follows from Lemma 5.1 that, when H satisfies (6), there is a commutative diagram

J^{1}ℝ^{k}|_{M} ⟶^{φ^{t}} J^{1}ℝ^{k}|_{M}
↓^{ρ}  ↓^{ρ}
J^{1}M ⟶^{ψ^{t}} J^{1}M

but note that φ^{t} is not uniquely determined by the restriction of H to J^{1}ℝ^{k}|_{M}.

We now return to the main problem and assume, from now on, that our contact isotopy ψ^{t} is generated by H. We define the variational family (P, Φ) as follows.

Let P = P_{ℝ^{k},M} be the space of paths

P = {c = (x, y): [0, 1]→T^{∗}ℝ^{k} | y(0) = 0, x(1) ∈ M}

in T^{∗}ℝ^{k}, and let π be the projection P → M given by c = (x, y) ↦ x(1). Define the action functional Φ = Φ_{H,M}: P → ℝ by

Φ(c) = ∫_{0}^{1} (⟨y(t), ẋ(t)⟩ − H(t, x(t), y(t), z(t))) dt

where z: [0, 1] → ℝ is the unique solution of the initial value problem

ż(t) = ⟨y(t), ẋ(t)⟩ − H(t, x(t), y(t), z(t)),  z(0) = 0.  (8)

Lemma 5.1 now implies the following:

Proposition 5.3 (P, Φ) is a variational family for the Legendrian submanifold ψ^{1}(M_{0}× {0}) ⊂ J^{1}M.

In particular, c ∈ P is a critical point of Φ if and only if (c, z) is a solution of Hamilton’s equations and c(1) is in T M^{⊥}, the conormal bundle of M.

Remark 5.4 The variational families (P_{M}, Φ_{H}), of Example 4.4, and (P, Φ), considered above, both generate the Legendrian submanifold ψ^{1}(M_{0} × {0}) ⊂ J^{1}M. These are related as follows: let c ∈ P be a fibre critical point of Φ, then \overline{c} ∈ P_{M}, given by composing c with the projection T^{∗}ℝ^{k}|_{M} → T^{∗}M, is a fibre critical point of Φ_{H}. Conversely any fibre critical point \overline{c} of Φ_{H} lifts uniquely to a fibre critical point c of Φ by using the contact isotopy φ^{t}.

5.2 The second variation

From the proof of Proposition 3.1 the differential of Φ is given by

dΦ(c)(γ) = ∫_{0}^{1} e^{∫_{t}^{1} −∂_{z}H dτ} (⟨η, ẋ − ∂_{y}H⟩ + ⟨ẏ + ∂_{x}H + y∂_{z}H, ξ⟩) dt
+ ⟨y(1), ξ(1)⟩

where γ ∈ T_{c}P = {γ = (ξ, η): [0, 1] → ℝ^{2k} | η(0) = 0, ξ(1) ∈ T_{x(1)}M}. In order to deal with the boundary term ⟨y(1), ξ(1)⟩ we now restrict Φ to the space

E = E_{ℝ^{k},M} = {c ∈ P | c(1) ∈ T M^{⊥}}

of paths satisfying Lagrangian boundary conditions. Before giving a formula for the second variation of Φ at a critical point we introduce some notation.

Let c ∈ P be a critical point of Φ and denote by z: [0, 1] → ℝ the corresponding solution of (8). Define, for reasons that will become apparent later, the matrix valued functions G_{ξξ}, G_{ξη}, G_{ηξ}, G_{ηη}: [0, 1] → ℝ^{k×k} and G_{ζ}: [0, 1] → ℝ by

\begin{aligned}G_{ξξ} & = ∂_{xx}^{2}H + ∂_{xz}^{2}Hy^{T} + y∂_{zx}^{2}H + ∂_{zz}^{2}Hyy^{T} \\ G_{ξη} & = ∂_{xy}^{2}H + y∂_{zy}^{2}H \\ G_{ηξ} & = ∂_{yx}^{2}H + ∂_{yz}^{2}Hy^{T} \\ G_{ηη} & = ∂_{yy}^{2}H \\ G_{ζ} & = ∂_{z}H. \end{aligned}  (9)

Here the derivatives of H are evaluated at (t, c(t), z(t)) ∈ [0, 1] × J^{1}ℝ^{k}. Note that G_{ξξ}(t) and G_{ηη}(t) are symmetric and that G_{ξη}(t) = G_{ηξ}(t)^{T}. Construct from these the symmetric matrix valued function S = S_{G}: [0, 1] → ℝ^{2k×2k} by

S(t) = (\begin{matrix}G_{ξξ}(t) & G_{ξη}(t) + \frac{1}{2}G_{ζ}(t)1l \\ G_{ηξ}(t) + \frac{1}{2}G_{ζ}(t)1l & G_{ηη}(t) \end{matrix}) .  (10)

Also, denote by J_{0} the 2k × 2k matrix

J_{0} = (\begin{matrix}0 & −1l \\ 1l & 0 \end{matrix})

and by C_{c} = C_{c,M} the tangent space of E at c:

C_{c} = {γ: [0, 1] → ℝ^{2k} | γ(0) ∈ Λ_{0}, γ(1) ∈ Λ_{1}} .

Here Λ_{0} = ℝ^{k} × {0} and Λ_{1} = T_{c(1)}(T M^{⊥}) are Lagrangian subspaces of ℝ^{2k}.

Proposition 5.5 The second differential of Φ at a critical point c is of the form

d^{2}Φ(c)(γ_{1}, γ_{2}) = ⟨A(c)γ_{1}, γ_{2}⟩_{L^{2}}

where γ_{1}, γ_{2} ∈ C_{c} and the second variation, A(c), is given by

A(c)γ = e^{∫_{t}^{1} −G_{ζ} dτ} (J_{0}γ̇ − Sγ + \frac{1}{2}G_{ζ}J_{0}γ) .  (11)

Proof. Fix a two parameter family of curves

c_{s_{1},s_{2}} = (x_{s_{1},s_{2}}, y_{s_{1},s_{2}}) ∈ E, s_{1}, s_{2} ∈ (−ε, ε)

with c_{0,0} = c and denote the derivatives by

ξ_{i,s_{1},s_{2}} = \frac{∂}{∂s_{i}} x_{s_{1},s_{2}}, η_{i,s_{1},s_{2}} = \frac{∂}{∂s_{i}} y_{s_{1},s_{2}}, ζ_{i,s_{1},s_{2}} = \frac{∂}{∂s_{i}} z_{s_{1},s_{2}}.

Here, of course, z_{s_{1},s_{2}} is given by solving (8) with x, y, z replaced by x_{s_{1},s_{2}}, y_{s_{1},s_{2}}, z_{s_{1},s_{2}}. Abbreviate ξ_{i} = ξ_{i,0,0}, η_{i} = η_{i,0,0}, ζ_{i} = ζ_{i,0,0} and set γ_{i} = (ξ_{i}, η_{i}), μ_{i} = (ξ_{i}, η_{i}, ζ_{i}). We now proceed with the proof of the proposition.

Differentiate the identity

ż_{s_{1},s_{2}} = ⟨y_{s_{1},s_{2}}, ẋ_{s_{1},s_{2}}⟩ − H(·, x_{s_{1},s_{2}}, y_{s_{1},s_{2}}, z_{s_{1},s_{2}})

with respect to s_{2} and s_{1} successively and then evaluate at s_{1} = s_{2} = 0 to obtain

∂_{s_{1}} ζ̇_{2} = ⟨∂_{s_{1}}η_{2}, ẋ⟩ + ⟨η_{2}, ξ̇_{1}⟩ + ⟨η_{1}, ξ̇_{2}⟩ + ⟨y, ∂_{s_{1}} ξ̇_{2}⟩
− d^{2}H_{t}(μ_{1}, μ_{2}) − ⟨∂_{x}H, ∂_{s_{1}}ξ_{2}⟩ − ⟨∂_{y}H, ∂_{s_{1}}η_{2}⟩ − ∂_{z}H∂_{s_{1}}ζ_{2}.

Using the fact that c is critical rewrite this as

ζ̇ = ⟨η_{2}, ξ̇_{1}⟩ + ⟨η_{1}, ξ̇_{2}⟩ + \frac{d}{dt}⟨y, ∂_{s_{1}}ξ_{2}⟩ + ⟨y∂_{z}H, ∂_{s_{1}}ξ_{2}⟩ − ∂_{z}Hζ − d^{2}H_{t}(μ_{1}, μ_{2})

where ζ = ∂_{s_{1}}ζ_{2} or more concisely, by using the substitution λ = ζ − ⟨y, ∂_{s_{1}}ξ_{2}⟩, as

\frac{d}{dt}λ = ⟨η_{2}, ξ̇_{1}⟩ + ⟨η_{1}, ξ̇_{2}⟩ − ∂_{z}Hλ − d^{2}H_{t}(μ_{1}, μ_{2}).

Since λ(0) = 0 the solution to this ODE evaluated at t = 1 is

λ(1) = ∫_{0}^{1} e^{∫_{t}^{1} −G_{ζ} dτ} (⟨η_{2}, ξ̇_{1}⟩ + ⟨η_{1}, ξ̇_{2}⟩ − d^{2}H_{t}(μ_{1}, μ_{2})) dt.

Also since 0 = ∂_{s_{1}}⟨y(1), ξ_{2}(1)⟩ = ⟨η_{1}(1), ξ_{2}(1)⟩ + ⟨y(1), ∂_{s_{1}}ξ_{2}(1)⟩ it follows that λ(1) = ζ(1) + ⟨η_{1}(1), ξ_{2}(1)⟩. Now partially integrate the second summand on the right hand side to obtain

ζ(1) = ∫_{0}^{1} e^{∫_{t}^{1} −G_{ζ} dτ} (⟨η_{2}, ξ̇_{1}⟩ − ⟨η̇_{1}, ξ_{2}⟩ − G_{ζ}⟨η_{1}, ξ_{2}⟩ − d^{2}H_{t}(μ_{1}, μ_{2})) dt.

Since c is critical observe, by Proposition 3.1, that

α(μ_{i}(t)) = ζ_{i}(t) − ⟨y(t), ξ_{i}(t)⟩ = 0

for all t ∈ [0, 1] thus, in particular,

d^{2}H_{t}(μ_{1}, μ_{2}) = ⟨G_{ξξ}ξ_{1}, ξ_{2}⟩ + ⟨G_{ξη}η_{1}, ξ_{2}⟩ + ⟨G_{ηξ}ξ_{1}, η_{2}⟩ + ⟨G_{ηη}η_{1}, η_{2}⟩.

The proposition now follows. □

Remark 5.6 A(c) extends to a self-adjoint operator on L^{2}([0, 1], ℝ^{2k}) with dense domain

W_{c,M}^{1,2} = {γ ∈ W^{1,2}([0, 1]; ℝ^{2k}) | γ(0) ∈ Λ_{0}, γ(1) ∈ Λ_{1}} .

In addition, A(c) is injective if and only if Ψ(1)Λ_{0} is transverse to Λ_{1} where Ψ(t) = Ψ_{c,H}(t) ∈ Sp(2k) is given by

Ψ̇ = −J_{0}SΨ, Ψ(0) = 1l.

In such a case we will say that c is a nondegenerate critical point of Φ.

Remark 5.7 The contact isotopy φ^{t}, generated by H, linearised along the path a = (c, z): [0, 1] → J^{1}ℝ^{k} and restricted to contact planes, gives rise to the path of linear conformal symplectomorphisms

dφ^{t}(a(0))|_{ξ_{a(0)}} : (ξ_{a(0)}, dα_{a(0)})→(ξ_{a(t)}, dα_{a(t)}).

This path may be naturally identified with the path e^{∫_{0}^{t} − \frac{1}{2} G_{ζ} dτ} Ψ_{c,H}, where Ψ_{c,H} is given in Remark 5.6, via the canonical symplectomorphisms

(ξ_{p}, dα_{p})→(ℝ^{2k}, ω_{0}): (ξ, η, ⟨y, ξ⟩) ↦ (ξ, η)

where p = (x, y, z).

Remark 5.8 Identify T^{∗}M with T M using the induced inner product. The tangent space to T M^{⊥} at p = (x, y) is given by

T_{p}(T M^{⊥}) = {(ξ, η) ∈ ℝ^{2k} : ξ ∈ T_{x}M, (dΠ(x)ξ)y + Π(x)η = 0}

where Π(x) ∈ ℝ^{k×k} denotes the orthogonal projection onto T_{x}M. Note that defining II_{p}: T_{x}M → ℝ by II_{p}(ξ) = \frac{1}{2}⟨(dΠ(x)ξ)y, ξ⟩ gives a quadratic form on T_{x}M, the second fundamental form of M at x along the normal vector y

Remark 5.9 Let G: [0, 1] × J^{1}ℝ^{k} → ℝ be the time-dependent Hamiltonian given by

G(t, ξ, η, ζ) = \frac{1}{2}⟨G_{ξξ}(t)ξ, ξ⟩ + ⟨G_{ηξ}(t)ξ, η⟩ + \frac{1}{2}⟨G_{ηη}(t)η, η⟩ + G_{ζ}(t)ζ.

We call such Hamiltonians, where G_{ξξ}(t), G_{ηξ}(t), G_{ηη}(t) ∈ ℝ^{k×k} with G_{ξξ}(t), G_{ηη}(t) symmetric and G_{ζ}(t) ∈ ℝ, quadratic, and define G_{ξη}(t) = G_{ηξ}(t)^{T}. Define the functional Φ_{c} = Φ_{c,H,M}: C_{c} → ℝ by

Φ_{c}(γ) = ∫_{0}^{1} (⟨η, ξ̇⟩ − G(t, ξ, η, ζ)) dt + II_{c(1)}(ξ(1)),

where ζ: [0, 1] → ℝ is the unique solution of the initial value problem

ζ̇(t) = ⟨η(t), ξ̇(t)⟩ − G(t, ξ(t), η(t), ζ(t)),  ζ(0) = 0.  (12)

Notice that if c is a nondegenerate critical point of Φ, then 0 is the unique critical point of Φ_{c}, also nondegenerate, and in this case the above proposition shows that the second variation of Φ at c agrees with the second variation of Φ_{c}.

Remark 5.10 A quadratic Hamiltonian G: [0, 1] × J^{1}ℝ^{k} → ℝ as above generates the contact isotopy

(\begin{matrix}ξ(t) \\ η(t) \end{matrix}) = e^{∫_{0}^{t} − \frac{1}{2} G_{ζ} dτ} Ψ_{G}(t) (\begin{matrix}ξ_{0} \\ η_{0} \end{matrix})
ζ(t) = e^{∫_{0}^{t} −G_{ζ} dτ} ζ_{0} + \frac{1}{2} ∫_{0}^{t} e^{∫_{s}^{t} −G_{ζ} dτ} (⟨G_{ηη}η, η⟩ − ⟨G_{ξξ}ξ, ξ⟩) ds

where Ψ_{G}: [0, 1] → Sp(2k) is given by

Ψ̇_{G} = −J_{0}S_{G}Ψ_{G},  Ψ_{G}(0) = 1l.  (13)

Conversely, given any smooth family of symplectic matrices Ψ: [0, 1] → Sp(2k) and any smooth positive function a: [0, 1] → ℝ_{>0} there exists a contact isotopy of the form

(\begin{matrix}ξ(t) \\ η(t) \end{matrix}) = a(t)Ψ(t) (\begin{matrix}ξ_{0} \\ η_{0} \end{matrix})
ζ(t) = a(t)^{2}ζ_{0} + Q(t, ξ_{0}, η_{0})

where Ψ = Ψ_{G} for some quadratic Hamiltonian G and Q(t, ·, ·) is quadratic. We call such contact isotopies pseudo-linear.

6 Discrete variational families

In this section we construct a family of finite dimensional variational families (P^{N}, Φ^{N}) for our Legendrian submanifold ψ^{1}(M_{0} × {0}) ⊂ J^{1}M. This is done in two stages. The first stage is to consider the case of a contactomorphism C^{1}-close to the identity. The second stage is the general case. Alternative constructions are given by Chekanov [5] and Théret [16]. The corresponding construction in the case of cotangent bundles was first carried out by Laudenbach and Sikorav [11].

6.1 Generating functions of type V

We generalise the notion of a generating function of type V, as defined in [12], to the contact setting. Let φ: J^{1}ℝ^{k} → J^{1}ℝ^{k} be a contactomorphism and denote it as follows:

(x_{0}, y_{0}, z_{0}) ↦ (x_{1}, y_{1}, z_{1}) = (u(x_{0}, y_{0}, z_{0}), v(x_{0}, y_{0}, z_{0}), w(x_{0}, y_{0}, z_{0})).

A generating function of type V for φ is a function V : ℝ^{2k+1} → ℝ such that

x_{1} − x_{0} = \frac{∂V}{∂y} (x_{0}, y_{1}, z_{0})
y_{1} − y_{0} = −\frac{∂V}{∂x} (x_{0}, y_{1}, z_{0}) − y_{0} \frac{∂V}{∂z} (x_{0}, y_{1}, z_{0})  (14)
z_{1} − z_{0} = ⟨y_{1}, x_{1} − x_{0}⟩ − V (x_{0}, y_{1}, z_{0})

if and only if (x_{1}, y_{1}, z_{1}) = φ(x_{0}, y_{0}, z_{0}). In this case, observe that φ has compact support (that is, φ is equal to the identity outside a compact set) if and only if V has.

Proposition 6.1 (i) Every contactomorphism φ: J^{1}ℝ^{k} → J^{1}ℝ^{k} which is sufficiently close to the identity in the C^{1}-topology admits a unique generating function of type V.

(ii) For each smooth function V : ℝ^{2k+1} → ℝ having compact support and sufficiently small first and second derivatives there exists a unique contactomorphism φ: J^{1}ℝ^{k} → J^{1}ℝ^{k} such that V is a generating function of type V for φ

Proof. For (i), the assumption on φ ensures that the map

ℝ^{2k+1} → ℝ^{2k+1} : (x_{0}, y_{0}, z_{0}) ↦ (x_{0}, v(x_{0}, y_{0}, z_{0}), z_{0})

has a global inverse. Thus there exists a map f: ℝ^{2k+1} → ℝ^{k} such that

y_{0} = f(x_{0}, v(x_{0}, y_{0}, z_{0}), z_{0}).

Also, since φ: (x_{0}, y_{0}, z_{0}) ↦ (x_{1}, y_{1}, z_{1}) is a contactomorphism, we have

dz_{1} − ∑y_{1i} dx_{1i} = e^{g} (dz_{0} − ∑y_{0i} dx_{0i})  (15)

for some function g: ℝ^{2k+1} → ℝ. Thus, using the notation set up earlier, the chain rule gives

dz_{1} − ∑y_{1i} dx_{1i} = ∑ (∂_{x_{0i}}w − ⟨y_{1}, ∂_{x_{0i}}u⟩) dx_{0i} + ∑ (∂_{y_{0i}}w − ⟨y_{1}, ∂_{y_{0i}}u⟩) dy_{0i} + (∂_{z_{0}}w − ⟨y_{1}, ∂_{z_{0}}u⟩) dz_{0}.

Comparing with (15) we find that

∂_{y_{0}}w − (∂_{y_{0}}u)^{T} y_{1} = 0,  (16)

and

∂_{x_{0}}w − (∂_{x_{0}}u)^{T} y_{1} = − (∂_{z_{0}}w − ⟨y_{1}, ∂_{z_{0}}u⟩) y_{0}.  (17)

We claim that V : ℝ^{2n+1} → ℝ defined by

V (x_{0}, y_{1}, z_{0}) := ⟨y_{1}, x_{1} − x_{0}⟩ − z_{1} + z_{0}
= ⟨y_{1}, u(x_{0}, f(x_{0}, y_{1}, z_{0}), z_{0}) − x_{0}⟩ − w(x_{0}, f(x_{0}, y_{1}, z_{0}), z_{0}) + z_{0} (18)

satisfies (14).

This claim is proved by a straightforward calculation, differentiating (18) and using the identities (16) and (17). The uniqueness of generating functions of type V is clear.

For (ii), let V : ℝ^{2k+1} → ℝ be a smooth function satisfying the hypotheses. We construct φ as follows. Let μ: ℝ^{2k+1} → ℝ^{2k+1} be the map defined by

(x_{0}, y_{1}, z_{0}) ↦ (x_{0}, (1 − ∂_{z_{0}}V (x_{0}, y_{1}, z_{0}))^{−1} (y_{1} + ∂_{x_{0}}V (x_{0}, y_{1}, z_{0})) , z_{0}) ,

and ν: ℝ^{2k+1} → ℝ^{2k+1} the map defined by

(x_{0}, y_{1}, z_{0}) ↦ (x_{0} + ∂_{y_{1}}V, y_{1}, z_{0} + ⟨y_{1}, ∂_{y_{1}}V ⟩ − V) .

The hypotheses on V ensure that μ and ν are diffeomorphisms. We define φ to be the composition ν ∘ μ^{−1}. That φ is a contactomorphism is seen directly as follows. Expand the formula for the contact 1-form α = dz_{1} −∑ y_{1i} dx_{1i}. After cancellations this becomes

dz_{1} − ∑y_{1i} dx_{1i} = (1 − ∂_{z_{0}}V) dz_{0} − ∑(∂_{x_{0i}}V + y_{1i}) dx_{0i}.

From the definition of the function y_{0} we have the equality

∂_{x_{0}}V + y_{1} = (1 − ∂_{z_{0}}V) y_{0}

hence φ is a contactomorphism as required. □

6.2 The pseudo-linear case

When φ is the time-t_{1} map of a contact isotopy generated by a quadratic Hamiltonian G the pair (ξ_{1}, η_{1}) is given in terms of (ξ_{0}, η_{0}) by

(\begin{matrix}ξ_{1} \\ η_{1} \end{matrix}) = aΨ (\begin{matrix}ξ_{0} \\ η_{0} \end{matrix})  (19)

where a > 0 and

Ψ = (\begin{matrix}A & B \\ C & D \end{matrix}) ∈ Sp(2k)

are as given in Remark 5.10. In this case the condition of φ being C^{1}-close to the identity is replaced by requirement that

det(D) ≠ 0  (20)

which holds for t_{1} > 0 sufficiently small. Again, we say a generating function ℝ^{2k+1} → ℝ: (ξ, η, ζ) ↦ W (ξ, η, ζ) is quadratic if it is quadratic in ξ and η and linear in ζ. A simple modification of the proof of Proposition 6.1 now gives:

Proposition 6.2 Every contactomorphism φ: J^{1}ℝ^{k} → J^{1}ℝ^{k} generated by a quadratic Hamiltonian with (20) holding admits a unique quadratic generating function of type V.

Conversely, for each function W = W (ξ, η, ζ) on ℝ^{2k+1}, quadratic in ξ and η and linear in ζ, satisfying

1 − ∂_{ζ}W ≠ 0,  det(1l + ∂_{ξη}^{2}W) ≠ 0,  (21)

there exists a unique pseudo-linear contactomorphism φ: J^{1}ℝ^{k} → J^{1}ℝ^{k} such that W is a generating function of type V for φ.

We now give an explicit formula for the quadratic generating function of the pseudolinear contactomorphism φ above. Observe from (19) that when condition (20) holds

ξ_{1} − ξ_{0} = (aA − 1l)ξ_{0} + aBη_{0}
= (aA − 1l)ξ_{0} + BD^{−1}η_{1} − aBD^{−1}Cξ_{0}
= (aA − aBD^{−1}C − 1l)ξ_{0} + BD^{−1}η_{1}
= (a(D^{−1})^{T} − 1l)ξ_{0} + BD^{−1}η_{1}.

Here the last equality follows from the fact that Ψ ∈ Sp(2k). Also note that

η_{1} − η_{0} = η_{1} − a^{−1}D^{−1}η_{1} + D^{−1}Cξ_{0}
= D^{−1}Cξ_{0} + (1l − aD^{−1})η_{1} + (a − a^{−1})D^{−1}η_{1}
= D^{−1}Cξ_{0} + (1l − aD^{−1})η_{1} + (a − a^{−1})D^{−1}(aCξ_{0} + aDη_{0})
= a^{2}D^{−1}Cξ_{0} + (1l − aD^{−1})η_{1} + (a^{2} − 1)η_{0}.

It follows that the generating function of type V for φ is given by

W (ξ_{0}, η_{1}, ζ_{0}) = −\frac{1}{2}⟨a^{2}D^{−1}Cξ_{0}, ξ_{0}⟩ + ⟨(a(D^{−1})^{T} − 1l)ξ_{0}, η_{1}⟩
+ \frac{1}{2}⟨BD^{−1}η_{1}, η_{1}⟩ + (1 − a^{2})ζ_{0}.  (22)

6.3 Discrete-time variational theory

We consider the general case. Recall that M is a closed manifold embedded ℝ^{k}, and H: [0, 1] × J^{1}ℝ^{k} → ℝ satisfies (6) for some function H: [0, 1] × J^{1}M → ℝ. It follows that H will not have compact support. Hence we multiply by a smooth cutoff function and assume, by abuse of notation, that H is equal to zero outside a large compact set containing ⋃_{t∈[0,1]} φ^{t}(M), where φ^{t} denotes the contact isotopy generated by H, and M the set M × {0} × {0} ⊂ J^{1}ℝ^{k}.

We now construct a family of finite dimensional variational families generating the Legendrian submanifold ψ^{1}(M_{0}×{0}), by discretising time. Pick an integer N and define

φ_{j}^{j+1} = φ^{\frac{j+1}{N}} ∘ (φ^{\frac{j}{N}} )^{−1}

for j = 0, …, N − 1. Then

φ^{1} = φ_{N−1}^{N} ∘ φ_{N−2}^{N−1} ∘ ⋯ ∘ φ_{0}^{1}

and for N sufficiently large each φ_{j}^{j+1} satisfies the hypotheses of Proposition 6.1 (i). Hence for each j there exists a function V_{j}: ℝ^{2k+1} → ℝ such that

x_{j+1} − x_{j} = \frac{∂V_{j}}{∂y}
y_{j+1} − y_{j} = −\frac{∂V_{j}}{∂x} − y_{j} \frac{∂V_{j}}{∂z}  (23)
z_{j+1} − z_{j} = ⟨y_{j+1}, x_{j+1} − x_{j}⟩ − V_{j}

if and only if (x_{j+1}, y_{j+1}, z_{j+1}) = φ_{j}^{j+1}(x_{j}, y_{j}, z_{j}).

Now define, by analogy with the continuous-time case, the space

P^{N} = P_{ℝ^{k},M}^{N} = {c = (x_{0}, …, x_{N}, y_{1}, …, y_{N}) ∈ ℝ^{(2N+1)k} | x_{N} ∈ M}

of discrete paths in ℝ^{2k}, and to each c ∈ P^{N} associate a sequence (z_{0}, …, z_{N}) given by the iterative rule

z_{0} = 0,  z_{j} = ⟨y_{j}, x_{j} − x_{j−1}⟩ − V_{j−1}(x_{j−1}, y_{j}, z_{j−1}) + z_{j−1}, j > 0.  (24)

Let π: P^{N} → M denote the projection c ↦ x_{N}. The discrete action functional Φ^{N} = Φ_{H,M}^{N}: P^{N} → ℝ can now be defined by

Φ^{N}(c) = z_{N} = ∑_{j=1}^{N} (⟨y_{j}, x_{j} − x_{j−1}⟩ − V_{j−1}(x_{j−1}, y_{j}, z_{j−1})).

Proposition 6.3 (P^{N}, Φ^{N}) is a variational family for the Legendrian submanifold ψ^{1}(M_{0} × {0}) ⊂ J^{1}M.

This follows from:

Lemma 6.4 There is a one-to-one correspondence between the set of fibre critical points of Φ^{N} and solutions of (23) (for j = 0, …, N − 1) with y_{0} = z_{0} = 0, x_{N} ∈ M.

Proof. The partial derivatives of Φ^{N} are

\frac{∂Φ^{N}}{∂x_{j}} = (1 − ∂_{z}V_{N−1}) ⋯ (1 − ∂_{z}V_{j+1}) (y_{j} − y_{j+1} − ∂_{x}V_{j} − y_{j}∂_{z}V_{j})
\frac{∂Φ^{N}}{∂y_{j+1}} = (1 − ∂_{z}V_{N−1}) ⋯ (1 − ∂_{z}V_{j+1}) (x_{j+1} − x_{j} − ∂_{y}V_{j})

for j = 0, …, N − 1 (on defining y_{0} = 0). Since 1 − ∂_{z}V_{l} is nonzero for each l, this proves the lemma. □

As before, c ∈ P^{N} is a critical point of Φ^{N} if and only if (c, z) is a solution of Hamilton’s discrete equations (23) and c_{N} = (x_{N}, y_{N}) ∈ T M^{⊥}. In particular, c is a critical point of Φ^{N} if and only if it is obtained by sampling a path c ∈ crit Φ.

Remark 6.5 In applications it will be useful to note the following. Let x = x_{N} and ξ = (x_{0}, …, x_{N−1}, y_{1}, …, y_{N}) denote the base and fibre coordinates respectively of P^{N} = M × ℝ^{2m} where m = Nk. Then Φ^{N} can be written in the form

Φ^{N}(x, ξ) = \frac{1}{2}⟨P ξ, ξ⟩ + W (x, ξ).

where P ∈ ℝ^{2m×2m} is a nondegenerate symmetric matrix of signature zero and W is given by

W (x, ξ) = ⟨x, y_{N}⟩ − ∑_{j=1}^{N} V_{j−1}(x_{j−1}, y_{j}, z_{j−1})

Also, the gradient of W with respect to ξ, ∂_{ξ}W, is bounded.

Remark 6.6 Suppose L ⊂ J^{1}M is a Legendrian submanifold given by a variational family (E = M × ℝ^{l}, S). Then ψ^{1}(L) is also given by a variational family. This is seen as follows.

Choose any function S̃: Ẽ → ℝ, where Ẽ = ℝ^{k} × ℝ^{l}, satisfying S̃|_{E} = S, and define the space

P̃^{N} = {(ξ, c) ∈ ℝ^{l} × ℝ^{(2N+1)k} | c ∈ P^{N}}

of augmented discrete paths. Now define the generalised discrete action functional Φ̃^{N}: P̃^{N} → ℝ by

Φ̃^{N}(ξ, c) = z_{N} = ∑_{j=1}^{N} (⟨y_{j}, x_{j} − x_{j−1}⟩ − V_{j−1}(x_{j−1}, y_{j}, z_{j−1})) + z_{0}

where

z_{0} = S̃(x_{0}, ξ), z_{j} = ⟨y_{j}, x_{j} − x_{j−1}⟩ − V_{j−1}(x_{j−1}, y_{j}, z_{j−1}) + z_{j−1}, j > 0.

It is now easy to check that (P̃^{N}, Φ̃^{N}) is a variational family for ψ^{1}(L).

Remark 6.7 An alternative and more explicit way of constructing a discrete action functional is by setting

V_{j}(x_{j}, y_{j+1}, z_{j}) = H(t_{j}, x_{j}, y_{j+1}, z_{j})(t_{j+1} − t_{j})

where t_{j} = j/N and N is sufficiently large so that the hypotheses of Proposition 6.1 (ii) hold for each V_{j}. A disadvantage of this method is that it does not generate the original Legendrian submanifold but one that is close to it. This is the approach used by Robbin and Salamon in the symplectic setting [13].

6.4 The second variation

We start with some notation. Let c ∈ P^{N} be a critical point of Φ^{N} and {z_{j}} the associated sequence given by the iterative rule (24). For j = 0, …, N − 1 define W_{j,ξξ}, W_{j,ξη}, W_{j,ηξ}, W_{j,ηη} ∈ ℝ^{k×k} and W_{j,ζ} ∈ ℝ by

W_{j,ξξ} = ∂_{xx}^{2}V_{j} + ∂_{xz}^{2}V_{j}y_{j}^{T} + y_{j}∂_{zx}^{2}V_{j} + ∂_{zz}^{2}V_{j}y_{j}y_{j}^{T}
W_{j,ξη} = ∂_{xy}^{2}V_{j} + y_{j}∂_{zy}^{2}V_{j}
W_{j,ηξ} = ∂_{yx}^{2}V_{j} + ∂_{yz}^{2}V_{j}y_{j}^{T}
W_{j,ηη} = ∂_{yy}^{2}V_{j}
W_{j,ζ} = ∂_{z}V_{j}

where V_{j} is evaluated at (x_{j}, y_{j+1}, z_{j}). Note that W_{j,ξξ}, W_{j,ηη} are symmetric and W_{j,ξη} = W_{j,ηξ}^{T}. Abbreviate α_{j}^{l} = (1 − ∂_{z}V_{l}) ⋯ (1 − ∂_{z}V_{j}) for l ≥ j, setting it to 1 otherwise. Denote by W_{c}^{N} = W_{c,M}^{N} the tangent space of P^{N} at c:

W_{c}^{N} = {γ = (ξ_{0}, …, ξ_{N}, η_{1}, …, η_{N}) ∈ ℝ^{(2N+1)k} | ξ_{N} ∈ T_{x_{N}} M} .

We now give a formula for the second variation of Φ^{N} at c. As before, Π(x) ∈ ℝ^{k×k} denotes the orthogonal projection onto T_{x}M.

Proposition 6.8 The second derivative of Φ^{N} at a critical point c is of the form

d^{2}Φ^{N}(c)(γ_{1}, γ_{2}) = ⟨A^{N}(c)γ_{1}, γ_{2}⟩

where γ_{1}, γ_{2} ∈ W_{c}^{N} and the second variation, A^{N}(c): W_{c}^{N} → W_{c}^{N}: (ξ, η) ↦ (u, v), is given by the expressions^{1}

u_{j} = α_{j+1}^{N−1}(η_{j} − η_{j+1} − W_{j,ξξ}ξ_{j} − W_{j,ξη}η_{j+1} − W_{j,ζ}η_{j})
v_{j+1} = α_{j+1}^{N−1}(ξ_{j+1} − ξ_{j} − W_{j,ηξ}ξ_{j} − W_{j,ηη}η_{j+1})
u_{N} = (dΠ(x_{N})ξ_{N})y_{N} + Π(x_{N})η_{N}.

Proof. Differentiate the formulae in the proof of Lemma 6.4. □

Remark 6.9 For j = 0, …, N − 1 let W_{j}: ℝ^{2k+1} → ℝ be the function defined by

W_{j}(ξ_{j}, η_{j+1}, ζ_{j}) = \frac{1}{2}⟨W_{j,ξξ}ξ_{j}, ξ_{j}⟩ + ⟨W_{j,ηξ}ξ_{j}, η_{j+1}⟩ + \frac{1}{2}⟨W_{j,ηη}η_{j+1}, η_{j+1}⟩ + W_{j,ζ}ζ_{j}.

Using these we may associate to the critical point c the function Φ_{c}^{N} = Φ_{c,H,M}^{N}: W_{c}^{N} → ℝ defined by

Φ_{c}^{N}(γ) = ∑_{j=1}^{N} (⟨η_{j}, ξ_{j} − ξ_{j−1}⟩ − W_{j−1}(ξ_{j−1}, η_{j}, ζ_{j−1})) + II_{c_{N}} (ξ_{N})

where ζ_{j} is given by the iterative rule

ζ_{0} = 0,  ζ_{j} = ⟨η_{j}, ξ_{j} − ξ_{j−1}⟩ − W_{j−1}(ξ_{j−1}, η_{j}, ζ_{j−1}) + ζ_{j−1}, j > 0  (25)

and where II_{c_{N}} denotes the second fundamental form. As in the continuous-time case, if c is a nondegenerate critical point of Φ^{N}, then 0 is the unique critical point of Φ_{c}^{N}, also nondegenerate, and in this case Proposition 6.8 implies that the second variation of Φ^{N} at c agrees with the second variation of Φ_{c}^{N}.

1Define η_{0} = 0.

Recall that in the continuous-time case given a critical point c of Φ we can construct a functional Φ_{c} whose second variation agrees with the second variation of Φ at c (see Remark 5.9). The following proposition shows how Φ_{c}^{N} and Φ_{c} are related.

Proposition 6.10 Φ_{c}^{N} is the discretisation of Φ_{c}.

Proof. Let (c, z): [0, 1] → J^{1}M be a solution of Hamilton’s equations beginning in L_{0}×{0} and (c^{N}, {z_{j}}) the corresponding solution of Hamilton’s discrete equations. Note that, by Remark 5.7, linearising Hamilton’s equations for H along (c, z) and restricting to contact planes, which are canonically identified with ℝ^{2k}, yields the equations

ξ̇ = G_{ηξ}ξ + G_{ηη}η
η̇ = −G_{ξξ}ξ − G_{ξη}η − G_{ζ}η.

Similarly, linearising Hamilton’s discrete equations along (c^{N}, {z_{j}}) and restricting to contact planes yields the equations

ξ_{j+1} − ξ_{j} = W_{j,ηξ}ξ_{j} + W_{j,ηη}η_{j+1}
η_{j+1} − η_{j} = −W_{j,ξξ}ξ_{j} − W_{j,ξη}η_{j+1} − W_{j,ζ}η_{j}.

It follows that the W_{j} define a family of generating functions of type V for the pseudolinear contact isotopy generated by G and the result now follows. □

Corollary 6.11 (P^{N}, Φ^{N}) is always transversal as a variational family.

Proof. We need to show that, given γ = (ξ_{0}, …, ξ_{N}, η_{1}, …, η_{N}) ∈ W_{c}^{N} satisfying ξ_{N} = 0, if d^{2}Φ^{N}(c)(γ, γ′) is 0 for all γ′ ∈ W_{c}^{N} then γ is identically 0. This is seen as follows. Suppose γ satisfies the hypotheses of the statement we are seeking to show, then

ξ_{j+1} − ξ_{j} = W_{j,ηξ}ξ_{j} + W_{j,ηη}η_{j+1}
η_{j+1} − η_{j} = −W_{j,ξξ}ξ_{j} − W_{j,ξη}η_{j+1} − W_{j,ζ}η_{j}

for j = 1, …, N − 1, and Π(x_{N})η_{N} = 0. Denoting by A_{j} the conformal symplectic matrix associated to W_{j}, it follows that

(\begin{matrix}ξ_{j+1} \\ η_{j+1} \end{matrix}) = A_{j} (\begin{matrix}ξ_{j} \\ η_{j} \end{matrix})

for j = 1, …, N −1, and that (ξ_{N}, η_{N}) is in the symplectic complement of the coisotropic subspace N_{c_{N}} = T_{c_{N}} (T^{∗}ℝ^{k}|_{M}). Since φ^{t} preserves J^{1}ℝ^{k}|_{M}, we have A_{j}N_{c_{j}} = N_{c_{j+1}}. It follows that (ξ_{0}, 0) is in the symplectic complement of N_{c_{0}} which implies that ξ_{0} = 0 and hence γ is identically 0, as required. □

Corollary 6.12 The second variation of Φ^{N} at a critical point c is nondegenerate if and only if Ψ_{c,H}Λ_{0} is transverse to Λ_{1} = T_{c_{N}} (T M^{⊥}).

Proof. This follows immediately from Corollaries 6.11 and 4.6 and Remark 5.7. □

Remark 6.13 If the variational family (E, S), given in Remark 6.6, is transversal, then the variational family (P̃^{N}, Φ̃^{N}) will also be transversal. This follows by arguing as in the proof of Corollary 6.11 and using Lemma 4.5.

7 The Maslov index

In this section we define the Maslov index for the points of intersection of ψ^{1}(M_{0} × {0}) and M_{0} × ℝ, assuming that the intersection is transversal. We begin by summarising the properties of the Maslov index for various spaces of paths that we will require. Proofs of these may be found in [14].

Let Λ: [a, b] → L(k) be a path of Lagrangian subspaces and let Z = (X, Y): [a, b] → L(ℝ^{k}, ℝ^{2k}) denote a choice of a frame for Λ. This means that Z(t): ℝ^{k} → ℝ^{2k} is an injective linear map such that

Λ(t) = Im Z(t), X(t)^{T} Y (t) = Y (t)^{T} X(t)

for all t. Also, let V ∈ L(k) be a fixed Lagrangian subspace. A crossing for the path Λ is a number t ∈ [a, b] such that Λ(t) intersects V nontrivially. At a crossing t the crossing form is defined to be the quadratic form

Γ(Λ, V, t): Λ(t) ∩ V →ℝ

given by

v ↦ ⟨X(t)u, Ẏ (t)u⟩ − ⟨Y (t)u, Ẋ(t)u⟩

where v = Z(t)u. A crossing t is said to be regular if Γ(Λ, V, t) is nonsingular. When the path Λ has only regular crossings the Maslov index, μ(Λ, V), is defined by

μ(Λ, V) = \frac{1}{2} sign Γ(Λ, V, a) + ∑_{a<t<b} sign Γ(Λ, V, t) + \frac{1}{2} sign Γ(Λ, V, b)

where the sum is taken over all crossings t. By perturbing, keeping endpoints fixed, this definition extends to give a well defined Maslov index for all paths. The integers k_{a} and k_{b} are defined by

k_{a} = dim Λ(a) ∩ V, k_{b} = dim Λ(b) ∩ V.

We now state some properties of the Maslov index. (Integrality) The integers μ, k_{a} and k_{b} are related by

μ + \frac{1}{2}(k_{a} − k_{b}) ∈ ℤ.

(Product) Under the natural identification of L(k) × L(k′) as a submanifold of L(k + k′)

μ(Λ ⊕ Λ′, V ⊕ V′) = μ(Λ, V) + μ(Λ′, V′).

(Localisation) For the path Λ given by the frame t ↦ (1l, A(t)) and V the subspace ℝ^{k} × {0} the Maslov index is given by

μ(Λ, V) = \frac{1}{2} sign A(b) − \frac{1}{2} sign A(a).

(Zero) If Λ(t) ∩ V has constant dimension for all t then μ(Λ, V) = 0.

For a pair of Lagrangian paths Λ, Λ′: [a, b] → L(k) the relative crossing form Γ(Λ, Λ′, t) on Λ(t) ∩ Λ(t)′ is defined by

Γ(Λ, Λ′, t) = Γ(Λ, Λ′(t), t) − Γ(Λ(t), Λ′, t).

When the pair Λ, Λ′ have only regular crossings the relative Maslov index, μ(Λ, Λ′), is defined by

μ(Λ, Λ′) = \frac{1}{2} sign Γ(Λ, Λ′, a) + ∑_{a<t<b} sign Γ(Λ, Λ′, t) + \frac{1}{2} sign Γ(Λ, Λ′, b).

The relative Maslov index has the following property. (Naturality) For a Lagrangian pair Λ, Λ′ and a symplectic path Ψ

μ(ΨΛ, ΨΛ′) = μ(Λ, Λ′).

Finally, for a path of symplectic matrices Ψ: [a, b] → Sp(2k) the Maslov index, μ(Ψ), is defined by^{2}

μ(Ψ) = μ(ΨV, V)

where V = ℝ^{k} × {0}.

The following property of the Maslov index for symplectic paths will be used. (Homotopy) Two symplectic paths Ψ, Ψ′ with Ψ(a) = Ψ′(a) and Ψ(b), Ψ′(b) ∈ Sp_{0}(2k) are homotopic within this class if and only if they have the same Maslov index, where Sp_{0}(2k) denote the set of matrices Ψ ∈ Sp(2k) such that ΨV transverse to V.

We now define the Maslov index for the points of intersection of ψ^{1}(M_{0} × {0}) and M_{0} × ℝ, or equivalently, the critical points of Φ_{H}.

Let \overline{c} ∈ P_{M} be a critical point of Φ_{H} and denote by c = (x, y) ∈ P its unique lift to a critical point of Φ (see Remark 5.4). Induced by c is the path of symplectic matrices

Ψ_{c,H}: [0, 1]→Sp(2k)

(see Remarks 5.6 and 5.7) and this reduces, via symplectic reduction of the coisotropic distribution T (T^{∗}ℝ^{k}|_{M}), to the path of symplectic linear transformations

Ψ_{\overline{c},H}(t): (T_{\overline{c}(0)}(T^{∗}M), −dλ_{can}_{\overline{c}(0)}) ⟶ (T_{\overline{c}(t)}(T^{∗}M), −dλ_{can}_{\overline{c}(t)}).

Now define the Maslov index, μ(\overline{c}, H), by

μ(\overline{c}, H) = μ(Ψ_{\overline{c},H} \overline{Λ}_{0}, \overline{Λ}_{1})  (26)

where \overline{Λ}_{0} = T_{x(0)}M and \overline{Λ}_{1} is the path obtained by reducing

Λ_{1}(t) = {(ξ, η) ∈ ℝ^{2k} : ξ ∈ T_{x(t)}M, Π(x(t))(dΠ(x(t))ξ)y(t) + Π(x(t))η = 0} . (27)

Here the right hand side of (26) is defined by choosing a unitary trivialisation of the vector bundle \overline{c}^{∗}T (T^{∗}M). By the naturality property this is independent of the trivialisation chosen. By the integrality property

μ(\overline{c}, H) ∈ \frac{n}{2} + ℤ.

If we assume that the embedding M ↪ ℝ^{k} is an isometry, then this definition is independent of the choice of embedding. Indeed, the induced Riemannian structure on T^{∗}M, via the embedding M ↪ ℝ^{k}, induces a splitting of T_{p}(T^{∗}M) into a horizontal and a vertical space:

2Note that this definition differs from the one given in [14] since here the Maslov index is defined with respect to the horizontal and there with respect to the vertical.

T_{p}(T^{∗}M) = H_{p} ⊕ V_{p},

given by

H_{p} = {(ξ, (dΠ(x)ξ)y) ∈ ℝ^{2k} : ξ ∈ T_{x}M}
V_{p} = {(0, η) ∈ ℝ^{2k} : η ∈ T_{x}M}

where p = (x, y). In this notation \overline{Λ}_{1}(t) is just the horizontal subspace H_{\overline{c}(t)}.

Lemma 7.1 Let \overline{c} ∈ P_{M} and c ∈ P be critical point of Φ_{H} and Φ respectively and related as above. Then there is an equality

μ(\overline{c}, H) = μ(Ψ_{c,H}Λ_{0}, Λ_{1}(1)) + \frac{1}{2} sign II_{c(1)}

where Λ_{1} : [0, 1] → L(k) is given by (27).

The proof of this lemma uses the following fact about symplectic reductions.

Lemma 7.2 Let Λ_{0}, Λ_{1} : [0, 1] → L(k) be a pair of Lagrangian paths and N ⊂ ℝ^{2k} a coisotropic subspace such that

Λ_{0}(t) ∩ N^{ω} = {0}, Λ_{1}(t) ⊂ N.

Then the Maslov index of the pair (Λ_{0}, Λ_{1}) agrees with the Maslov index of the reduced pair (\overline{Λ}_{0}, \overline{Λ}_{1}):

μ(Λ_{0}, Λ_{1}) = μ(\overline{Λ}_{0}, \overline{Λ}_{1}).

Proof. By choosing a basis for the isotropic subspace N^{ω} and then extending it to a basis of the Lagrangian subspace Λ_{1}(t) we can assume that

N = ℝ^{n} × {0} × ℝ^{n} × ℝ^{k−n}
Λ_{1} = ℝ^{n} × {0} × {0} × ℝ^{k−n}

for all t. Hence it is sufficient to check that the assertion of the proposition holds when only Λ_{0} is allowed to vary and Λ_{1} and N are as above. Since Λ_{0}(t) ∩ N^{ω} = {0} it follows that we may choose a Lagrangian frame for Λ_{0} of the form Z = (X, Y): [0, 1] → L(ℝ^{k}, ℝ^{2k}) where

X = (\begin{matrix}X & A \\ 0 & 1l \end{matrix}) , Y = (\begin{matrix}\overline{Y} & B \\ C & D \end{matrix})

and thus a Lagrangian frame for \overline{Λ}_{0} is given by Z = (X, \overline{Y}): [0, 1] → L(ℝ^{n}, ℝ^{2n}). We compute the crossing form for the pair (Λ_{0}, Λ_{1}).

A crossing for (Λ_{0}, Λ_{1}) is a number t ∈ [0, 1] such that there is a u = (u′, 0) ∈ ℝ^{k} such that \overline{Y} (t)u′ = 0, that is, if and only if t is a crossing for (\overline{Λ}_{0}, \overline{Λ}_{1}). At a crossing t the crossing form is given by

Γ(Λ_{0}, Λ_{1}, t)(v) = ⟨X(t)u, Ẏ (t)u⟩ − ⟨Y (t)u, Ẋ(t)u⟩
= ⟨X(t)u′, ˙ \overline{Y} (t)u′⟩
= Γ(\overline{Λ}_{0}, \overline{Λ}_{1}, t)(v′)

where v = Z(t)u, v′ = Z(t)u′. This proves the lemma. □

Proof of Lemma 7.1. In view of Lemma 7.2 it is sufficient to show that

μ(Ψ_{c,H}Λ_{0}, Λ_{1}) = μ(Ψ_{c,H}Λ_{0}, Λ_{1}(1)) + \frac{1}{2} sign II_{c(1)}.

Let Φ: [0, 1] → Sp(2k) be a path of symplectic matrices satisfying

Λ_{1}(t) = Φ(t)Λ_{1}(1)

and abbreviate Ψ = Ψ_{c,H}. Then

μ(ΨΛ_{0}, Λ_{1}) = μ(ΨΛ_{0}, ΦΛ_{1}(1))
= μ(Φ^{−1}ΨΛ_{0}, Λ_{1}(1))
= μ(Φ(1)^{−1}ΨΛ_{0}, Λ_{1}(1)) + μ(Φ^{−1}Ψ(0)Λ_{0}, Λ_{1}(1))
= μ(ΨΛ_{0}, Λ_{1}(1)) + μ(Λ_{0}, Λ_{1}).

By multiplying by a path of unitary matrices if necessary we may assume that

Λ_{1}(t) = {(ξ, 0, −B(t)ξ, η) ∈ ℝ^{2k} : ξ ∈ ℝ^{n}, η ∈ ℝ^{k−n}}

where B: [0, 1] → ℝ^{n×n} is a path of symmetric matrices with B(0) = 0. Now by the product and localisation properties of the Maslov index

μ(Λ_{0}, Λ_{1}) = \frac{1}{2} sign B(1) = \frac{1}{2} sign II_{c(1)}.

This proves the lemma. □

8 The signature identity

In this section we generalise the signature theorem of Robbin and Salamon in [13] to the case of nondegenerate critical points of the action functional on the 1-jet bundle of a closed manifold.

Theorem 8.1 Let c ∈ P be a nondegenerate critical point of Φ, \overline{c} its projection to P_{M} and c^{N} ∈ P^{N} the corresponding critical point of Φ^{N}. Then, for sufficiently large N, the signature of the second variation of Φ^{N} at c^{N} is given by

sign A^{N}(c^{N}) = 2μ(\overline{c}, H).

This strengthens the theorem of Viterbo [17], which states that, in the symplectic case, the difference of the signature of the second variation of Φ^{N} at two critical points is independent of N, but at the cost of having to take N sufficiently large.

Below we prove a result which implies this theorem.

Let G: [0, 1] × J^{1}ℝ^{k} → ℝ be a quadratic Hamiltonian and denote by Ψ_{G}: [0, 1] → Sp(2k) the associated path of symplectic matrices given by (13). Also, let F ⊂ ℝ^{k} be a linear subspace and B: F → F a linear transformation such that ⟨Bξ, ξ′⟩ = ⟨ξ, Bξ′⟩ for all ξ, ξ′ ∈ F. Define the Lagrangian subspace

Λ_{F,B} = {(ξ, η) ∈ ℝ^{2k} : ξ ∈ F, Bξ + Π_{F} η}

Where Π_{F} ∈ ℝ^{k×k} denotes the orthogonal projection onto F. Now define the function Φ_{G,F,B}^{N}: W_{F}^{N} → ℝ by

Φ_{G,F,B}^{N}(γ) = ∑_{j=1}^{N} (⟨η_{j}, ξ_{j} − ξ_{j−1}⟩ − W_{j−1}(ξ_{j−1}, η_{j}, ζ_{j−1})) + \frac{1}{2}⟨Bξ_{N}, ξ_{N}⟩

where

W_{F}^{N} = {γ = (ξ_{0}, …, ξ_{N}, η_{1}, …, η_{N}) ∈ ℝ^{(2N+1)k} | ξ_{N} ∈ F }

and where the W_{j} are computed using (22), and the ζ_{j} using (25). Denote by A_{G,F,B}^{N} the second variation of Φ_{G,F,B}^{N}.

In view of Proposition 6.10 and Lemma 7.1, Theorem 8.1 now follows from:

Theorem 8.2 Let G, F and B be as above and assume that Ψ_{G}(1)Λ_{0} is transverse to Λ_{F,B}. Then, for N sufficiently large, the signature of the second variation of Φ_{G,F,B}^{N} is given by

sign A_{G,F,B}^{N} = 2μ(Ψ_{G}Λ_{0}, Λ_{F,B}) + sign B.

The proof we give is based on that given by Robbin and Salamon [13] in the case where F = ℝ^{k}, B = 0 and G is independent of z.

Proof. Without loss of generality assume that

Λ_{F,B} = {(ξ′, 0, −Bξ′, η′′) ∈ ℝ^{n} × ℝ^{k−n} × ℝ^{n} × ℝ^{k−n} = ℝ^{2k} : ξ′ ∈ ℝ^{n}, η′′ ∈ ℝ^{k−n}}

where dim F = n and where B ∈ ℝ^{n×n} is symmetric.

The proof of the theorem now proceeds in three steps.

Step 1. The theorem holds in the case

G(t, ξ′, ξ′′, η′, η′′, ζ) = \frac{1}{2}⟨G_{ξ′ξ′}(t)ξ′, ξ′⟩.

In this case the path of symplectic matrices associated to G is given by

Ψ_{G}(t) = ( \begin{matrix}1l & 0 & 0 & 0 \\ 0 & 1l & 0 & 0 \\ C(t) & 0 & 1l & 0 \\ 0 & 0 & 0 & 1l \end{matrix} )

where

C(t) = ∫_{0}^{t} −G_{ξ′ξ′}(τ) dτ.

Thus, from the product and the localisation properties of the Maslov index,

μ(Ψ_{G}Λ_{0}, Λ_{F,B}) = \frac{1}{2} sign(C(1) + B) − \frac{1}{2} sign B.

Also, from (22),

W_{j}(ξ_{j}, η_{j+1}, ζ_{j}) = \frac{1}{2}⟨W_{j,ξ′ξ′}ξ′_{j}, ξ′_{j}⟩,

and defining

C^{j} = ∑_{i=0}^{j−1} −W_{i,ξ′ξ′}

for j = 1, …, N, it is easy to see that C^{N} = C(1). Hence the second variation is given by

⟨A_{G,F,B}^{N}γ, γ⟩ = 2 ∑_{j=1}^{N}⟨η_{j}, ξ_{j} − ξ_{j−1}⟩ − ∑_{j=0}^{N−1}⟨W_{j,ξ′ξ′}ξ′_{j}, ξ′_{j}⟩ + ⟨Bξ′_{N}, ξ′_{N}⟩.

Now define the following system of coordinates on W_{F}^{N} = ℝ^{2Nk+n}:

u′_{j} = ξ′_{j} − ξ′_{j−1}
u′′_{j} = ξ′′_{j} − ξ′′_{j−1}
v′_{j} = η′_{j} − \frac{1}{2}C^{j}(ξ′_{j} + ξ′_{j−1})
v′′_{j} = η′′_{j}
w = ξ′_{N}

and denote u = (u′_{1}, u′′_{1}, …, u′_{N}, u′′_{N}), v = (v′_{1}, v′′_{1}, …, v′_{N}, v′′_{N}). In these coordinates the L^{2}-inner product of u and v is given by

⟨u, v⟩ = ∑_{j=1}^{N}⟨u′_{j}, v′_{j}⟩ + ∑_{j=1}^{N}⟨u′′_{j}, v′′_{j}⟩

= ∑_{j=1}^{N}⟨ξ′_{j} − ξ′_{j−1}, η′_{j}⟩ − \frac{1}{2} ∑_{j=1}^{N}⟨ξ′_{j} − ξ′_{j−1}, C^{j}(ξ′_{j} + ξ′_{j−1})⟩

+ ∑_{j=1}^{N}⟨ξ′′_{j} − ξ′′_{j−1}, η′′_{j}⟩

= ∑_{j=1}^{N}⟨ξ_{j} − ξ_{j−1}, η_{j}⟩ + \frac{1}{2} ∑_{j=1}^{N}⟨C^{j}ξ′_{j−1}, ξ′_{j−1}⟩ − \frac{1}{2} ∑_{j=1}^{N}⟨C^{j}ξ′_{j}, ξ′_{j}⟩

= ∑_{j=1}^{N}⟨ξ_{j} − ξ_{j−1}, η_{j}⟩ + \frac{1}{2} ∑_{j=0}^{N−1}⟨C^{j+1}ξ′_{j}, ξ′_{j}⟩ − \frac{1}{2} ∑_{j=0}^{N}⟨C^{j}ξ′_{j}, ξ′_{j}⟩

= ∑_{j=1}^{N}⟨ξ_{j} − ξ_{j−1}, η_{j}⟩ + \frac{1}{2} ∑_{j=0}^{N−1}⟨(C^{j+1} − C^{j})ξ′_{j}, ξ′_{j}⟩ − \frac{1}{2}⟨C^{N}ξ′_{N}, ξ′_{N}⟩

= ∑_{j=1}^{N}⟨ξ_{j} − ξ_{j−1}, η_{j}⟩ − \frac{1}{2} ∑_{j=0}^{N−1}⟨W_{j,ξ′ξ′}ξ′_{j}, ξ′_{j}⟩ − \frac{1}{2}⟨C^{N}ξ′_{N}, ξ′_{N}⟩

= \frac{1}{2}⟨A_{G,F}^{N} γ, γ⟩ − \frac{1}{2}⟨(C^{N} + B)w, w⟩.

Thus the second variation satisfies

\frac{1}{2}⟨A_{G,F,B}^{N}γ, γ⟩ = ⟨u, v⟩ + \frac{1}{2}⟨(C^{N} + B)w, w⟩

and hence

sign A_{G,F,B}^{N} = sign (C^{N} + B)
= sign(C(1) + B)
= 2μ(Ψ_{G}Λ_{0}, Λ_{F,B}) + sign B.

Step 2. The theorem holds for quadratic Hamiltonians independent of ζ.

Let G be a ζ-independent Hamiltonian with Ψ_{G}(1)Λ_{0} transverse to Λ_{F,B} and suppose, for a given value of k′, that G′: ℝ^{2k′} → ℝ is any symplectic shear of the form

G′(ξ′, η′) = \frac{1}{2}⟨G′_{ξ′ξ′}ξ′, ξ′⟩

where G′_{ξ′ξ′} is a nondegenerate symmetric matrix of signature zero. (It follows that k′ is necessarily even.) From Step 1 it follows that

sign A_{G′,ℝ^{k′} ,0}^{N} = 2μ(Ψ_{G′}Λ′_{0}, Λ′_{0}) = 0

where Λ′_{0} = ℝ^{k′} × {0}. Now define G_{0} = G ⊕ G′: [0, 1] × J^{1}ℝ^{k+k′} → ℝ by

G_{0}(t, ξ, ξ′, η, η′, ζ) = G(t, ξ, η, ζ) + G′(ξ′, η′).

By additivity of the signature

sign A_{G_{0},F ⊕ℝ^{k′} ,B⊕0}^{N} = sign A_{G,F,B}^{N}

and by the product property of the Maslov index

μ(Ψ_{G_{0}}Λ_{0} ⊕ Λ′_{0}, Λ_{F,B} ⊕ Λ′_{0}) = μ(Ψ_{G}Λ_{0}, Λ_{F,B}).

Hence it suffices to prove the theorem for G_{0} with F replaced by F̃ = F ⊕ ℝ^{k′} and B by B̃ = B ⊕ 0.

Now, for k′ sufficiently large, let G_{1} be a symplectic shear of the form considered in Step 1, with F replaced by F̃, satisfying

μ(Ψ_{G_{1}} Λ̃_{0}, Λ_{F̃,B̃}) = μ(Ψ_{G_{0}} Λ̃_{0}, Λ_{F̃,B̃})

where Λ̃_{0} = Λ_{0} ⊕ Λ′_{0}. This exists by the localisation and integrality properties of the Maslov index. Also, choose a symplectic matrix Φ ∈ Sp(2k +2k′) such that Λ_{F̃,B̃} = ΦΛ̃_{0}. Then

μ(Φ^{−1}Ψ_{G_{1}} Λ̃_{0}, Λ̃_{0}) = μ(Φ^{−1}Ψ_{G_{0}} Λ̃_{0}, Λ̃_{0})

and the symplectic path Φ^{−1}Ψ_{G_{i}} satisfies

Φ^{−1}Ψ_{G_{i}}(0) = Φ^{−1}, Φ^{−1}Ψ_{G_{i}}(1)Λ̃_{0} ∩ Λ̃_{0} = {0}.

Thus by the homotopy property of the Maslov index there exists a homotopy of symplectic paths Φ^{−1}Ψ_{λ}: [0, 1] → Sp(2k + 2k′) between Φ^{−1}Ψ_{G_{0}} and Φ^{−1}Ψ_{G_{1}} and within the same class. From this we can construct a quadratic Hamiltonian G_{λ} such that Ψ_{λ} = Ψ_{G_{λ}}.

Now choose N sufficiently large such that for each λ the function Φ_{G_{λ},F̃,B̃}^{N} is defined. Then

sign A_{G_{0},F̃,B̃}^{N} = sign A_{G_{1},F̃,B̃}^{N}
= 2μ(Ψ_{G_{1}} Λ̃_{0}, Λ_{F̃,B̃}) + sign B̃
= 2μ(Ψ_{G_{0}} Λ̃_{0}, Λ_{F̃,B̃}) + sign B̃.

Here the first equality follows from Corollary 6.12.

Step 3. The general case.

To the quadratic Hamiltonian

G(t, ξ, η, ζ) = \frac{1}{2}⟨G_{ξξ}(t)ξ, ξ⟩ + ⟨G_{ηξ}(t)ξ, η⟩ + \frac{1}{2}⟨G_{ηη}(t)η, η⟩ + G_{ζ}(t)ζ

associate the quadratic ζ-independent Hamiltonian

G′(t, ξ, η, ζ) = \frac{1}{2}⟨G_{ξξ}(t)ξ, ξ⟩ + ⟨(G_{ηξ}(t) + \frac{1}{2}G_{ζ}(t)1l)ξ, η⟩ + \frac{1}{2}⟨G_{ηη}(t)η, η⟩.

Now consider the homotopy G_{λ} = λG′ + (1 − λ)G between G and G′. Notice that Ψ_{G_{λ}} = Ψ_{G} as symplectic paths for all λ. Step 3 now follows from Corollary 6.12. This completes the proof of the theorem. □

9 Proof of the generalised Morse inequalities

Recall from Remark 6.5 that the discrete action functional Φ^{N}: M × ℝ^{2m} → ℝ can be written in the form

Φ^{N}(x, ξ) = \frac{1}{2}⟨P ξ, ξ⟩ + W (x, ξ)

where P ∈ ℝ^{2m×2m} is a nondegenerate symmetric matrix of signature zero and where the gradient of W in the direction of the fibre, ∂_{ξ}W, is bounded. We prove the generalised Morse inequalities by studying the critical points of Φ^{N}. As M × ℝ^{2m} is noncompact stabilised Morse theory will be used; see Conley [6].

In order to study the critical points of Φ^{N} we consider the (negative) gradient flow. This is defined by the equations

\frac{dξ}{ds} = −P ξ − \frac{∂W}{∂ξ} (x, ξ), \frac{dx}{ds} = −\frac{∂W}{∂x} (x, ξ).

Now induced by P there is a splitting

ℝ^{2m} = E^{−} ⊕ E^{+}

into the negative and positive eigenspaces. It follows that, for some δ > 0,

⟨P ξ^{−}, ξ^{−}⟩ ≤ −δ|ξ^{−}|^{2}
⟨P ξ^{+}, ξ^{+}⟩ ≥ δ|ξ^{+}|^{2}

for ξ^{−} ∈ E^{−}, ξ^{+} ∈ E^{+}. Thus we can find constants ε > 0 and R > 0 such that

\frac{d}{ds}|ξ^{−}|^{2} ≥ ε if |ξ^{−}| ≥ R
\frac{d}{ds}|ξ^{+}|^{2} ≤ −ε if |ξ^{+}| ≥ R.

Indeed, for |ξ^{−}| ≥ R and R sufficiently large,

\frac{d}{ds} \frac{|ξ^{−}|^{2}}{2} = ⟨ξ^{−}, −P ξ^{−} − ∂_{ξ}W ⟩ ≥ δ|ξ^{−}|^{2} − sup |∂_{ξ}W | · |ξ^{−}| ≥ ε,

and similarly for |ξ^{+}|^{2} ≥ R. Thus an isolating block, in the sense of Conley, for the compact invariant set Λ of all bounded orbits of the gradient flow, is given by

N = {(x, ξ^{−} + ξ^{+}) : ξ^{±} ∈ E^{±}, |ξ^{±}| ≤ R}

with exit set

L = {(x, ξ^{−} + ξ^{+}) ∈ N : |ξ^{−}| = R}

for R sufficiently large.

We use the following notation

b_{k}(Λ) = dim H_{k}(N, L)
b_{k}(M) = dim H_{k}(M)
c_{k}(Λ) = # {c ∈ Λ | dΦ^{N}(c) = 0, ind_{Φ^{N}} (c) = k}
p_{k}(Λ) = # {c ∈ Λ | dΦ^{N}(c) = 0, μ(c, H) = k}

where ind_{Φ^{N}} (c) denotes the Morse index of Φ^{N} at c and is defined to be the number of negative eigenvalues of the Hessian d^{2}Φ^{N}(c). Here the numbers b_{k}(Λ) are known as the Conley–Betti numbers. These are related to the c_{k}(Λ) by the Morse inequalities.

Theorem 9.1 (Morse inequalities) For k = 0, …, n + 2m

c_{k}(Λ) − c_{k−1}(Λ) + ⋯ ± c_{0}(Λ) ≥ b_{k}(Λ) − b_{k−1}(Λ) + ⋯ ± b_{0}(Λ)

with equality holding for k = n + 2m.

These inequalities are proved in [12].

The Conley–Betti numbers are related to the Betti numbers of M by the Thom isomorphism. Specifically, denote by E^{m} m-dimensional Euclidean space, by B^{m} and S^{m−1} the closed unit ball and unit sphere respectively in E^{m}, and abbreviate E_{0}^{m} = E^{m} \ {0}. Then by homotopy

(M × B^{m} × B^{m}, M × ∂B^{m} × B^{m}) ≃ (M × B^{m}, M × S^{m−1}) ≃ (M × E^{m}, M × E_{0}^{m}).

Thus

H_{k}(N, L) = H_{k}(M × E^{m}, M × E_{0}^{m}) = H_{k−m}(M)

where the last equality is the Thom isomorphism. In other words

b_{k}(Λ) = b_{k−m}(M).  (28)

Also by Theorem 8.1

c_{k} = p_{\frac{n}{2} +m−k}.  (29)

The proof of Theorem 1.1, the generalised Morse inequalities, is now completed by putting together (28), (29) and the Morse inequalities.

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