Smooth integrability of implicit differential systems

Takuo FUKUDA and Stanislaw JANECZKO ^{∗}^{†}

1 Introduction

If P is a smooth manifold and V : P → T P is a vector field on P, i.e. a section of the tangent bundle π_{1} : T P → P, then the local integrability of V is a characteristic property, especially for smooth sections V. Traditionally the local integrability of a vector field V (not necessary continuous) on P is defined as an existence at each point p of the domain of V a C^{1}-curve γ : (−ε, ε) → P (for some ε > 0), such that γ(0) = p and \frac{∂γ}{∂t} (t) = V (γ(t)) for all t ∈ (−ε, ε). The curve γ is called an integral curve of V with initial value p. We may assume that the existence of such γ for V is not an exceptional property , and that for each p ∈ P there exists a neighbourhood U of p and ε > 0 such that the mapping U × (−ε, ε) ∋ (p̄, t) → γp̄(t), γp̄(0) = p̄, is defined and at least continuous. For smooth, compactly supported V this mapping forms a one parameter group of transformations of P.

When we replace V by a smooth embedded manifold M into T P, then the mentioned integrability property becomes rather the restrictive exceptional property for M, especially in the critical point set of the projection π_{1} |_{M}: M → P. In general, such systems are called implicit differential systems (equations) of order 1, and were studied by several authors: In connection to the non continuous integrability property by [18, 6]. The local equivalences of singularities of such systems were studies (among others) by [19, 3, 4, 17]. The Hamiltonian implicit case which is especially addressed to mathematical physics was introduced to mathematics by P.A.M. Dirac [5], investigated also by [13, 2, 9].

The local smooth integrability property is important for understanding the local dynamics of the system and the structure of the induced transformation groups. However the integrability property is not a usual property in the implicit case. Then a natural question which arises, is to characterize those implicit systems M which are integrable, i.e. through each point of M there is a local integrable curve passing through. Another natural question is to understand the reason for non integrability - the local topological structure of the strata of non integrable points.

The purpose of this paper is threefold: First we provide necessary and sufficient conditions of integrability for various classes of implicit systems. Second we formulate and investigate the problem of integrability for implicit Hamiltonian systems as a specialized class of implicit systems, and Third we search for the generic peculiarities of non integrable points.

∗AMS(2000) subject classification. Primary: 57R45, Secondary: 58C35, 70H05, 34A26

†Supported by KBN grant Nr 2 P03A 020 17

In Sections 2,3 and 4 we formulate the integrability problem, consider the two dimensional fold and cusp cases and prove the first part of the main results of the paper: the smooth integrability necessary and sufficient conditions for the generic (in the sense of J. Mather) differential systems (Theorem 3.1 and Theorem 3.2). In Section 5 the special case of Morin corank 1 stable singularities is considered in details and in Section 6 the crucial examples and counter examples of various integrability and non integrability situations are constructed. Passing to real analytic category, in Section 7, we extend J. Mather’s results weakening his transversality condition to the irreducibility of determinants. This result is formulated in Theorem 7.1 and Theorem 7.2. Along the road to this result we prove several applicable properties of prime ideals having the property of zeroes. Specialization of our differential systems to the Hamiltonian ones, which was a part of the motivation for this paper, was done in Section 8. In this case we applied the main results of the previous sections and obtained the necessary and sufficient conditions for integrability of implicit Hamiltonian systems. Here we develop the computable formula for the integrability check in case of P.A.M. Dirac generalized Hamiltonian systems. Finally in Section 9 the integrability on submanifolds is investigated (Theorem 9.1, Theorem 9.2) and Section 10 is devoted to the non-integrable points of differential systems. We prove here (Theorem 10.1, Theorem 10.2) that the non-integrable points are isolated generically in the sense of J.C. Tougeron.

2 The integrability problem

Let M be a smooth submanifold of T R^{n}, M ⊂ T R^{n} ≡ R^{n} × R^{n}, dimM = m. M is considered as a first order differential system.

Definition 2.1 A point (p, q) ∈ M ⊂ T R^{n} ≡ R^{n} × R^{n} is called an integrable point of M if there exists a smooth curve γ : (−ε, ε) → R^{n} such that

γ(0) = p, γ′(0) = q and (γ(t), γ′(t)) ∈ M for all t ∈ (−ε, ε).

The manifold M is called an integrable manifold if M consists only of integrable points.

Now we have a natural question.

Problem 2.1 Find a necessary and sufficient condition for a manifold M ⊂ T R^{n} to be integrable.

We immediately find a necessary condition as follows. Let π_{1} : T R^{n} = R^{n} ×R^{n} → R^{n} be the projection to the first factor:

π_{1} : R^{n} × R^{n} → R^{n}, π_{1}(p, q) = p, tangent bundle projection.

Let (p, q) ∈ M be an integrable point of M. Then from the definition there exists a smooth curve γ : (−ε, ε) → R^{n} such that

γ(0) = p, γ′(0) = q and (γ(t), γ′(t)) ∈ M.

Then we have

q = γ′(0) = \frac{d(π_{1}(γ, γ′))}{dt}(0) = d(π_{1} | M)_{(p,q)}(γ′(0), γ′′(0))

∈ d(π_{1} | M)_{(p,q)}(T_{(p,q)}M).

Thus an immediate necessary condition for a point (p, q) ∈ M to be integrable is that

q ∈ d(π_{1} | M)_{(p,q)}(T_{(p,q)}M).

In what follows we will call this condition N-integrability condition.

We can ask whether this condition is also a sufficient condition for a submanifold M to be integrable. Although the answer for this question is negative, there is a wide class of submanifolds of T R^{n} for which the N-integrability condition is also sufficient. An example of the submanifold M for which N-integrability condition is fulfilled but M is not integrable is given in Section 6, (see Example 6.1).

To come closer to our problem we explain some sufficient conditions for a submanifold M of T R^{n} to be integrable. One of them plays an important role in the proof that manifolds with corank 1 stable singularities are integrable. At first we have the following two immediate Lemmas.

Lemma 2.1 If a submanifold M ⊂ T R^{n} is a finite union of integrable submanifolds M_{i} ⊂ T R^{n};

M = ∪_{i}M_{i}, M_{i} integrable,

then M is integrable.

Lemma 2.2 Let M ⊂ T R^{n} be a submanifold satisfying N-integrability condition.

1) If π_{1} |_{M}: M → R^{n} is either an embedding or a submersion, then M is integrable.

2) If π_{1}(M) is a submanifold of R^{n} and π_{1} |_{M}: M → π_{1}(M) is a submersion, then M is integrable.

Since our problem is a local one, we may suppose that the manifold M under consideration is the image of an embedding of R^{m}.

M = (f, g)(R^{m}), (f, g) : R^{m} → T R^{n} = R^{n} × R^{n} is an embedding.

Let Jf(x) denote the Jacobian matrix of f = (f_{1}, …, f_{n}) at x = (x_{1}, …, x_{m}) ∈ R^{m}.

Jf(x) = ( \begin{matrix}\frac{∂f_{1}}{∂x_{1}} (x) & & & … & & \frac{∂f_{1}}{∂x_{m}} (x) \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ \frac{∂f_{n}}{∂x_{1}} (x) & & & … & & \frac{∂f_{n}}{∂x_{m}} (x) \end{matrix} )

Lemma 2.3 Let M = (f, g)(R^{m}), where

(f, g) : R^{m} → T R^{n} = R^{n} × R^{n} is an embedding.

Let M satisfies the N-integrability condition. If there exist smooth local functions a_{1}(x_{1}, …, x_{m}), …, a_{m}(x_{1}, …, x_{m}) such that

( \begin{matrix}g_{1}(x) \\ g_{2}(x) \\ … \\ g_{n}(x) \end{matrix} ) = ( \begin{matrix}\frac{∂f_{1}}{∂x_{1}} (x) & & & … & & \frac{∂f_{1}}{∂x_{m}} (x) \\ \frac{∂f_{2}}{∂x_{1}} (x) & & & … & & \frac{∂f_{2}}{∂x_{m}} (x) \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ \frac{∂f_{n}}{∂x_{1}} (x) & & & … & & \frac{∂f_{n}}{∂x_{m}} (x) \end{matrix} ) ( \begin{matrix}a_{1}(x) \\ a_{2}(x) \\ … \\ a_{m}(x) \end{matrix} ) ,  (1)

then M is integrable.

Proof. Take any point (p_{0}, q_{0}) = (f, g)(x_{0}) ∈ M. Consider the vector field

ξ(x) = a_{1}(x) (\frac{∂}{∂x_{1}} )_{x} + a_{2}(x) (\frac{∂}{∂x_{2}} )_{x} + … + a_{m}(x) (\frac{∂}{∂x_{m}} )_{x}

defined on R^{m}. Consider an integral curve

γ̃ : (−ε, ε) → R^{m} of ξ such that γ̃(0) = x_{0}

and consider the curve γ : (−ε, ε) → R^{n}, γ(t) = f(γ̃(t)). Then the point (p_{0}, q_{0}) = (f, g)(x_{0}) ∈ M is an integrable point of M with the curve γ(t). This can be seen as follows. Since

\frac{dγ̃}{dt} (t) = a(γ̃(t)) = (a_{1}(γ̃(t)), …, a_{m}(γ̃(t)),

we have

\frac{dγ}{dt} (t) = \frac{d(f ∘ γ̃)(t)}{dt} = Jf(γ̃(t)) · \frac{dγ̃}{dt} (t)

= ( \begin{matrix}\frac{∂f_{1}}{∂x_{1}} (γ̃(t)) & & … & & & \frac{∂f_{1}}{∂x_{m}} (γ̃(t)) \\ \frac{∂f_{2}}{∂x_{1}} (γ̃(t)) & & … & & & \frac{∂f_{2}}{∂x_{m}} (γ̃(t)) \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ \frac{∂f_{n}}{∂x_{1}} (γ̃(t)) & & … & & & \frac{∂f_{n}}{∂x_{m}} (γ̃(t)) \end{matrix} ) ( \begin{matrix}a_{1}(γ̃(t)) \\ a_{2}(γ̃(t)) \\ … \\ a_{m}(γ̃(t)) \end{matrix} ) = ( \begin{matrix}g_{1}(γ̃(t)) \\ g_{2}(γ̃(t)) \\ … \\ g_{n}(γ̃(t)) \end{matrix} ) .

Hence we have

(γ(t), γ′(t)) = (f(γ̃(t)), g(γ̃(t)) ∈ M = (f, g)(R^{m})
and (γ(0), γ′(0)) = (p_{0}, q_{0}).

Thus (p_{0}, q_{0}) is an integrable point of M. Q.E.D.

Corollary 2.1 Let f : R^{m} → R^{n} be a smooth mapping with

rankdf_{x} ≥ \frac{m}{2}, ∀x ∈ R^{m}.

Then there exist (always) a smooth mapping g : R^{m} → R^{n} such that (f, g) : R^{m} → R^{n} × R^{n} is an embedding and M = (f, g)(R^{m}) is an integrable manifold.

Proof. Since we consider the local problem and since

r = rankdf_{x} : T_{x}(R^{m}) → T_{f(x)}R^{n} ≥ \frac{m}{2}, ∀x ∈ R^{m},

we may suppose that f is of the form

f(x_{1}, …, x_{m}) = (x_{1}, …, x_{r}, f_{r+1}(x_{1}, …, x_{m}), …, f_{n}(x_{1}, …, x_{m})).

Let a_{1}(x), …, a_{n}(x) be the smooth functions chosen in the following form

a_{1}(x) = x_{r+1}, …, a_{m−r}(x) = x_{m} and a_{m−r+1}(x), …, a_{n}(x) being arbitrary.

And consider a mapping g = (g_{1}, …, g_{n}) : R^{m} → R^{n} defined by equation (1). Then (f, g) : R^{m} → R^{n} ×R^{n} is an embedding and from Lemma 2.3 we see that M = (f, g)(R^{m}) is an integrable manifold. Q.E.D.

Remark 2.1 Now we came to the natural question: Does some pathology of the critical point set on M imply the non integrability ? To some extent the unswer is positive. We may formulate the following supposition: if f : R^{m} → R^{n} is stable, then for any g : R^{m} → R^{n} such that (f, g) : R^{m} → R^{n} × R^{n} is an embedding and M = (f, g)(R^{m}) satisfies Nintegrability condition, M has to be integrable. Therefore if this conjecture is true, it implies that if M is not integrable, then f is not stable, and therefore π_{1} |_{M}: M → R^{n} is not a stable map. Here let us remind that the two maps f : R^{m} → R^{n} and π_{1} |_{M}: M → R^{n} are right equivalent in J.Mather’s sense, since (f, g) : R^{m} → M = (f, g)(R^{m}) is a diffeomorphism. Thus the singular point set is not the one of a stable map. On the other hand, the Corollary 2.1 says that if

rank(dπ_{1} |_{M}) : T_{(p,q)}(M) → T_{p}(R^{n}) ≥ \frac{m}{2},

then no matter how much pathological its singularity is, with a good partner g, M = (f, g)(R^{m}) becomes integrable.

To complete our sufficient condition formulated in Lemma 2.3 we present Example 6.2, in Section 6 showing that the condition of existence of smooth solution of the linear equation in Lemma 2.3 is not a necessary condition for integrability.

Now continuing the practical search for sufficient conditions we show that the Nintegrability condition is also sufficient for fold and cusp singularities of π_{1} |_{M} . At first we introduce the natural equivalency of equidimensional submanifolds in T R^{n}

Definition 2.2 Let M, N be smooth submanifolds of T R^{n}. We say that M and N are equivalent if there exists a diffeomorphism h : R^{n} → R^{n} and so its tangent bundle lifting H(p, q) = (h(p), Jh(p)q), where Jh(p) is the Jacobian matrix of h at p ∈ R^{n}, such that N = H(M).

Lemma 2.4 Let M and N be smooth submanifolds of T R^{n} = R^{n} × R^{n}. If M and N are equivalent, then N is integrable if and only if so is M.

Proof. Since the Jacobian matrix JH(p, q) has the form

JH(p, q) = ( \begin{matrix}Jh(p) & O \\ O & Jh(p) \end{matrix} ) ,

H is a diffeomorphism. Suppose that (p, q) ∈ M is an integrable point of M. Then there exists a smooth curve

γ : (−ε, ε) → R^{n} such that γ(0) = p, γ′(0) = q, (γ(t), (γ′(t)) ∈ M.

Consider a curve α : (−ε, ε) → R^{n} given by α(t) = h(γ(t)). Now we can see that the point (p′, q′) = H(p, q) ∈ N = H(M) is an integrable point of N with the curve α(t) as follows.

α(0) = h(γ(0)) = h(p) = p′, α′(0) = Jh(p)γ′(0) = Jh(p)q = q′,
α′(t) = Jh(γ(t))γ′(t).

Therefore

(α(t), α′(t)) = (h(γ(t)), Jh(γ(t))(γ′(t))) = H(γ(t), γ′(t)) ∈ H(M) = N.

The converse is the same. Q.E.D.

Proposition 2.1 Let M ⊂ T R^{2} = R^{2} × R^{2} be a manifold satisfying N-integrability condition. Let (p, q) ∈ M be a cusp or fold singularity of π_{1} |_{M}: M → R^{2}. Then (p, q) is an integrable point of M.

Proof. It is enough to prove this proposition by lemma 2.4 for the normal form of cusp singularity. The proof for the fold singularity goes in the similar way and it is easier. From Lemma 2.4, we may suppose the manifold M under consideration is the image of a map (f, g)) : R^{2} → R^{2} × R^{2} such that

1) f has the normal form of cusp singularity

f_{1}(x, y) = x, f_{2}(x, y) = y^{3} + xy,

2) g = (g_{1}, g_{2}) : R^{2} → R^{2} is any smooth map such that (f, g) : R^{2} → R^{2} × R^{2} is an embedding and

3) M = (f, g)(R^{2}) satisfies N-integrability condition.

We are going to prove that M = (f, g)(R^{2}) is integrable.

Since regular points of π_{1} |_{M}: M → R^{2} are integrable from Lemma 2.2, it suffices to consider singular points of π_{1} |_{M}: M → R^{2}. Let (p, q) = (p_{1}, p_{2}, q_{1}, q_{2}) = (f(x, y), g(x, y)) be a singular point of π_{1} |_{M}: M → R^{2}. Then we have

Jf(x, y) = ( \begin{matrix}1 & 0 \\ y & 3y^{2} + x \end{matrix} ) = ( \begin{matrix}1 & 0 \\ y & 0 \end{matrix} ) .

Then N-integrability condition is equivalent to the condition that

( \begin{matrix}g_{1}(x, y) \\ g_{2}(x, y) \end{matrix} ) ∈ Jf(x, y)(R^{2}) = ( \begin{matrix}1 & 0 \\ y & 0 \end{matrix} ) (R^{2}) = R ( \begin{matrix}1 \\ y \end{matrix} ) .

Therefore if g_{1}(x, y) = 0, we have also g_{1}(x, y) = 0 and q = (0, 0). We know that any point (p, q) with q = (0, 0) is always an integrable point of M.

Suppose now that g_{1}(0, 0) = c ≠ 0. Then g_{1}(x, y) has the form

g_{1}(x, y) = c + α(x, y) with α(0, 0) = 0.

This implies that g_{2}(x, y) coincides with yg_{1}(x, y) on the singular point set Σ(f) of f:

g_{2}(x, y) |_{Σ(f)}≡ yg_{1}(x, y) |_{Σ(f)} .

Let E_{x,y} denote the ring of smooth function-germs of two variables x, y , m_{x,y} denotes its maximal ideal and E_{x,y} < β > the ideal in E_{x,y} generated by β ∈ m_{x,y}. With these notations we have

g_{1}(x, y) = c + α(x, y) with α ∈ m_{x,y}.

Now the condition g_{2}(x, y) |_{Σ(f)}≡ yg_{1}(x, y) |_{Σ(f)} . implies (g_{2}(x, y) − yg_{1}(x, y)) |_{Σ(f)}≡ 0. But this implies that

g_{2}(x, y) − yg_{1}(x, y) ∈ E_{x,y} < 3y^{2} + x >,

because

Σ(f) = {(x, y) ∈ R^{2} | det Jf(x, y) = 3y^{2} + x = 0}

and the generator 3y^{2} +x of the ideal E_{x,y} < 3y^{2} +x > is irreducible. Therefore g_{2}(x, y)− yg_{1}(x, y) has the form

g_{2}(x, y) − yg_{1}(x, y) = (3y^{2} + x)β(x, y) for some β ∈ E_{x,y}.

Recalling the form

g_{1}(x, y) = c + α(x, y), c ≠ 0, α ∈ m_{x,y},

the map (f, g)) : R^{2} → R^{2} × R^{2} turns out to have the form

(f, g)(x, y) = (x, y^{3} + xy, c + α(x, y), cy + yα(xy) + (3y^{2} + x)β(x, y)),

J(f, g)(0, 0) = ( \begin{matrix}1 & 0 \\ 0 & 0 \\ ⋆ & ⋆ \\ ⋆ & c \end{matrix} ) , c ≠ 0.

Thus (f, g)) : R^{2} → R^{2} × R^{2} is an embedding.

Now let us observe the form of g(x, y);

( \begin{matrix}g_{1}(x, y) \\ g_{2}(x, y) \end{matrix} ) = ( \begin{matrix}c + α(x, y) \\ y(c + α(x, y)) + (3y^{2} + x)β(x, y) \end{matrix} )

= ( \begin{matrix}1 & 0 \\ y & 3y^{2} + x \end{matrix} ) ( \begin{matrix}c + α(x, y) \\ β(x, y) \end{matrix} ) .

Thus the functions

a_{1}(x, y) = c + α(x, y), a_{2}(x, y) = β(x, y)

satisfy the condition in Lemma 2.3 and M = (f, g)(R^{2}) is integrable. Q.E.D.

3 Generic manifolds are integrable

Now we prove that the N-integrability condition is also sufficient for a class of manifolds, which we call the generic manifolds and which is much wider than the class of manifolds M ⊂ T R^{n} = R^{n} × R^{n} such that the projections π_{1} |_{M}: M → R^{n} are C^{∞}−stable.

Let J^{1}(M, R^{n}) be the 1 jet bundle of 1 jets of maps of M into R^{n}, and let Σ_{r}(M, R^{n}) be the subset of all 1 jets j^{1}f(x) ∈ J^{1}(M, R^{n}) such that the rank of the differential df_{x} : T_{x}(M) → T_{f(x)}R^{n} at x ∈ M is r. Then Σ_{r}(M, R^{n}) is a submanifold of J^{1}(M, R^{n}) of codimension (n − r) × (m − r), where m = dimM.

Definition 3.1 (following J. Mather [12]) A manifold M ⊂ T R^{n} = R^{n} × R^{n} is called a generic submanifold of T R^{n} if the 1 jet section j^{1}(π_{1} |_{M}) : M → J^{1}(M, R^{n}) is transversal to all Σ_{r}(M, R^{n}), r = 0, 1, …, min(m, n).

Now we can formulate our first main theorem.

Theorem 3.1 A generic submanifold M of T R^{n} is integrable if and only if it satisfies N-integrability condition. Thus N-integrability condition holds sufficient for generic submanifolds of T R^{n}.

Remark 3.1 Remark that generic submanifolds form a very wide class in the set of all submanifolds of T R^{n}. This can be seen as follows. Since our problem is a local one, we may suppose that the manifold M under consideration is the image of an embedding of R^{m} into T R^{n}:

M = (f, g)(R^{m}), (f, g) : R^{m} → T R^{n} = R^{n} × R^{n} an embedding.

Then, the 1 jet section j^{1}(π_{1} |_{M}) : M → J^{1}(M, R^{n}) is transversal to Σ_{r}(M, R^{n}) if and only if the 1 jet section j^{1}f : R^{m} → J^{1}(R^{m}, R^{n}) is transversal to Σ_{r}(R^{m}, R^{n}). By Thom’s transversality theorem, the set of smooth maps f : R^{m} → R^{n} such that j^{1}f : R^{m} → J^{1}(R^{m}, R^{n}) is transversal to all Σ_{r}(R^{m}, R^{n}), r = 0, 1, …, min(m, n) form an open dense set in the set of smooth maps of R^{m} into R^{n}. Thus generic submanifolds form a very wide class in the set of all submanifolds of T R^{n}.

Since our problem is a local one, let us consider the set germ

(M, (p, q)) = ((f, g)(R^{m}), (f(0), g(0)))

at (p, q) ∈ M of a manifold M ⊂ T R^{n}. Now we are ready to formulate the second main theorem concerning the local characterization of integrable manifolds.

Theorem 3.2 Let M = (f, g)(R^{m}) ⊂ T R^{n} be a submanifold of T R^{n}.

1) The germ (M, (p, q)) = ((f, g)(R^{m}), (f(0), g(0))) is generic if and only if j^{1}f : R^{m} → J^{1}(R^{m}, R^{n}) is transversal to Σ_{r}(R^{m}, R^{n}) at 0 ∈ R^{m}, where r is the rank of the Jacobian matrix Jf(0) at 0 ∈ R^{m}.

2) Let (M, (p, q)) be the germ at (p, q) of a generic submanifold M. Then, (M, (p, q)) is integrable if and only if there exists a smooth map germ a : (R^{m}, 0) → R^{m} such that the equation (1) is fulfilled.

Now we reformulate our problem to have it in more accessible terms.

Definition 3.2 A smooth map g : R^{m} → R^{n} is said to be tangent to a smooth map f : R^{m} → R^{n} if the pair (f, g) : R^{m} → T R^{n} = R^{n} × R^{n} forms an embedding and the linear equation (1) has a solution (a_{1}(x), a_{2}(x), …, a_{n}(x)) for every point x ∈ R^{m}.

In this situation, N-integrability condition becomes exactly the tangency of the smooth map g : R^{m} → R^{n} to the smooth map f : R^{m} → R^{n}, defining the embedding (f, g) : R^{m} → T R^{n} = R^{n} × R^{n} of M.

Now we came to the following problem.

Problem 3.1 Find conditions to be posed on a smooth map f : R^{m} → R^{n} so that the image M = (f, g)(R^{m}) be an integrable manifold for every smooth map g : R^{m} → R^{n} tangent to f.

Also we can reformulate this Problem in other words as follows: Find conditions to be posed on a smooth map f : R^{m} → R^{n} so that the linear equation (1) has a smooth solution (a_{1}(x), a_{2}(x), …, a_{m}(x)) for every smooth map g : R^{m} → R^{n} tangent to f.

The result of [12] concerning solutions of linear equation helps to give a good answer to this problem. In the rest of this section, we will follow the notation from [12]. Let D^{k} denote the unit disk in R^{k}: D^{k} = {x ∈ R^{k} :∥ x ∥≤ 1}. Consider an n × m matrix M(x) of smooth functions on D^{k}:

M(x) = ( \begin{matrix}m_{11}(x) & & m_{12}(x) & & … & & & m_{1m}(x) \\ m_{21}(x) & & m_{22}(x) & & … & & & m_{2m}(x) \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ m_{n1}(x) & & m_{n2}(x) & & … & & & m_{nm}(x) \end{matrix} ) ,

and a column vector

g(x) = ( g_{1}(x) ; ⋮ ; g_{n}(x) ) ,

where m_{ij}(x) and g_{i} are smooth functions defined on D^{k}. Consider the linear equation

M(x)a = g(x),  (2)

where a is a column vector of length m.

Let E(n, m) denote the space of n × m matrices of real numbers. Let Σ_{r} denote the subset of E(n, m) consisting of matrices of rank r. Then Σ_{r} is a submanifold of codimension (n − r) × (m − r) in E(n, m).

Definition 3.3 An n×m matrix M(x) of smooth functions on D^{k} is said to be generic if M : D^{k} → E(n, m) and M : ∂D^{k} → E(n, m) are both transversal to all Σ_{r}, r = 0, 1, …, min(n, m).

J. Mather’s Theorem 1:([12]) If M(x) is generic and the equation M(x)a = g(x) has a solution for every x ∈ D^{k}, then Equation (2) has a smooth solution a(x).

Let E(n, m; k) denote the set of n × m matrices M(x) of smooth functions on D^{k}. Then E(n, m; k) may be identified with the set of smooth maps of D^{k} into E(n, m). We endow E(n, m; k) with the C^{1} topology. Then we have.

J. Mather’s Theorem 2:([12]) Generic matrices form an open dense set in E(n, m; k).

In the proof of J. Mather’s Theorem 1, the following local version of the theorem is also proved.

J. Mather’s Theorem 3:([12]) Let M : (R^{k}, 0) → E(n, m) be smooth map-germ and let g : (R^{k}, 0) → R^{n} be a smooth map-germ such that the linear equation (2) has a solution a(x) for every x ∈ R^{k} close to the origin 0 ∈ R^{k}. If the map-germ M : (R^{k}, 0) → E(n, m) is transversal to all Σ_{r}, r = 0, 1, …, min(n, m) at every point x ∈ R^{k} close to the origin 0 ∈ R^{k}, then the equation (2) has a local smooth solution a(x) defined in a neighbourhood of the origin 0 ∈ R^{k}.

4 Proof of Main Theorems

Since our problem is local, we use the notation of germs. Let f : R^{m} → R^{n} be a smooth mapping. From Lemma 2.4 we may suppose that f(0) = 0 without loss of generality. Denote by f : (R^{m}, 0) → (R^{n}, 0) its map-germ at 0 ∈ R^{m}. Consider the map

Jf : R^{m} → E(n, m) given by Jf(x) = the jacobian matrix at x ∈ R^{m}

and its germ

Jf : (R^{m}, 0) → E(n, m) at 0 ∈ R^{m}.

Now the statement of the Theorem 3.1 becomes of the following form.

Theorem 3.1: Let the rank of Jf(0) be equal to r. If Jf : (R^{m}, 0) → E(n, m) is transversal to Σ_{r} at 0 ∈ R^{m}, then, for any smooth map-germ g : (R^{m}, 0) → R^{n} tangent to f : (R^{m}, 0) → (R^{n}, 0), the germ of the manifold

M = (f, g)(R^{m}) ⊂ T R^{n} = R^{n} × R^{n}

at (0, g(0)) is integrable.

Now we have immediately.

Corollary 4.1 Let f : (R^{m}, 0) → (R^{n}, 0) be a C^{∞} stable map-germ. Then for any smooth map-germ g : (R^{m}, 0) → R^{n} tangent to f : (R^{m}, 0) → (R^{n}, 0), the manifold germ (M = (f, g)(R^{m}), (f(0) = 0, g(0))) is integrable.

Proof. Since stable maps are transversal to every A−orbit in the jet spaces (cf. [11]), so they are also transversal to Σ_{r}. Thus from the Theorem 3.1, the germ of the manifold M = (f, g)(R^{m}) at (0, g(0)) is integrable. Q.E.D.

The main theorem 3.1 follows from J. Mather’s Theorem 3 and the previous Lemma 2.3. Proof of Theorem 3.1 Let the rank of Jf(0) be equal to r. Suppose that Jf : (R^{m}, 0) → E(n, m) is transversal to Σ_{r} at 0 ∈ R^{m}. Since the stratification

{Σ_{i}}_{i=0}^{min{m,n}}

of E(n, m) satisfies Whitney’s condition a, this implies that Jf : (R^{m}, 0) → E(n, m) is transversal all Σ_{i}(R^{m}, R^{n}) at every point near the origin 0 ∈ R^{m}. Thus from J. Mather’s Theorem 3, for any smooth map-germ g : (R^{m}, 0) → R^{n} tangent to f : (R^{m}, 0) → (R^{n}, 0), the linear equation Jf(x)a = g(x) has a smooth solution a(x) defined in a neighbourhood of the origin 0 ∈ R^{m}. Therefore, from Lemma 2.3, the germ of the manifold (M = (f, g)(R^{m}), (0, g(0))) is integrable. Q.E.D.

Proof of Theorem 3.2 Proof of 1) The ”only if” part is trivial. Now suppose that j^{1}f : R^{m} → J^{1}(R^{m}, R^{n}) is transversal to Σ_{r}(R^{m}, R^{n}) at 0 ∈ R^{m}. Since the stratification

{Σ_{i}(R^{m}, R^{n})}_{i=0}^{min(m,n)}

of J^{1}(R^{m}, R^{n}) satisfies Whitney’s condition (a) (cf. [8]), this implies that j^{1}f : R^{m} → J^{1}(R^{m}, R^{n}) is transversal to all Σ_{i}(R^{m}, R^{n}) at every point near the origin 0 ∈ R^{m}. Thus the germ (M, (p, q)) is generic.

Proof of 2) This follows from the proof of the Theorem 3.1. Q.E.D.

5 Integrability for Morin corank 1 stable singularities

The class of generic manifolds which we considered in the previous section includes also those with stable singularities. However J. Mather’s theorem used in the proof of the Theorems 3.1 and 3.2 does not explain directly what is the real reason of the integrability. In this section we prove that the N-integrability condition is also sufficient for manifolds M with corank 1 stable singularities of the projection π_{1} |_{M} and show the main reason for the integrability being the fact that the Jacobian ideal of the singularity is generated by irreducible functions. Morin singularities with m = n Let f : (R^{m}, 0) → (R^{m}, 0) be a stable smooth map-germ with corank 1. Then it is right-left equivalent to the following canonical form:

f(x_{1}, …, x_{m−1}, y) = (x_{1}, …, x_{m−1}, f_{m}(x, y)), f_{m}(x, y) = y^{k+1} + ∑_{i=1}^{k−1} x_{i}y^{i}.

Its Jacobian matrix has the form

Jf(x, y) = ( \begin{matrix}1 & 0 & ⋯ & ⋯ & ⋯ & 0 & 0 \\ 0 & ... & & & O & & ⋮ \\ & & ... & & & ⋮ & ⋮ \\ ⋮ & & & ... & & & ⋮ \\ & O & & & ... & 0 & ⋮ \\ 0 & & ⋯ & & 0 & 1 & 0 \\ y & ⋯ & y^{k−1} & 0 & ⋯ & 0 & ∂f_{m}/∂y \end{matrix} ) ,

\frac{∂f_{m}}{∂y} (x, y) = x_{1} + ∑_{i=2}^{k−1} ix_{i}y^{i−1}(k + 1)y^{k}.

Proposition 5.1 Let f : (R^{m}, 0) → (R^{m}, 0) be a smooth stable corank 1 map-germ. Let g : (R^{m}, 0) → (R^{m}, 0) be a smooth map-germ such that

1) (f, g) : (R^{m}, 0) → T R^{m} = R^{m} × R^{m} is an embedding,

2) the image manifold-germ M = (f, g)((R^{m}, 0)) satisfies N-integrability condition, then M is integrable.

Proof.N-integrability condition is equivalent to the condition that the linear equation

Jf(x, y) ( a_{1} ; ⋮ ; a_{m} ) = ( g_{1}(x, y) ; ⋮ ; g_{m}(x, y) )

has a not necessarily smooth solution a(x, y) for every (x, y). If the above equation has a smooth solution, then from the Lemma 2.3 M is integrable.

At a point (x, y) where detJf(x, y) ≠ 0, the above equation has a local smooth solution and the point (x, y) is an integrable point. From the form of the equation Jf(x, y)a(x, y) = g(x, y)

( \begin{matrix}1 & 0 & ⋯ & ⋯ & ⋯ & 0 & 0 \\ 0 & ... & & & O & & ⋮ \\ & & ... & & & ⋮ & ⋮ \\ ⋮ & & & ... & & & ⋮ \\ & O & & & ... & 0 & ⋮ \\ 0 & & ⋯ & & 0 & 1 & 0 \\ y & ⋯ & y^{k−1} & 0 & ⋯ & 0 & ∂f_{m}/∂y \end{matrix} ) ( a_{1} ; a_{2} ; ⋮ ; a_{m−1} ; a_{m} ) = ( g_{1}(x, y) ; g_{2}(x, y) ; ⋮ ; g_{m−1}(x, y) ; g_{m}(x, y) ) ,

a_{1}(x, y), …, a_{m−1}(x, y) are uniquely determined and smooth:

a_{1}(x, y) = g_{1}(x, y), …, a_{m−1}(x, y) = g_{m−1}(x, y).

At points (x, y) where detJf(x, y) = 0, we have

g_{m}(x, y) = ∑_{i=1}^{k−1} y^{i}a_{i}(x, y) = ∑_{i=1}^{k−1} y^{i}g_{i}(x, y),

hence

(g_{m}(x, y) − ∑_{i=1}^{k−1} y^{i}g_{i}(x, y)) |_{{Jf(x,y)=0}}≡ 0.

Since

\frac{∂f_{m}}{∂y} (x, y) = x_{1} + ∑_{i=2}^{k−1} ix_{i}y^{i−1}(k + 1)y^{k}

is irreducible, (g_{m}(x, y) − ∑_{i=1}^{k−1} y^{i}g_{i}(x, y)) is contained in the ideal E_{xy} < ∂f_{m}/∂y >, where E_{xy} is the ring of smooth function-germs at 0 ∈ R^{m}. Therefore there exists a smooth function α(x, y) ∈ E_{xy} such that

g_{m}(x, y) − ∑_{i=1}^{k−1} y^{i}g_{i}(x, y) = α(x, y)\frac{∂f_{n}}{∂y} (x, y).

Thus we have

( \begin{matrix}1 & 0 & ⋯ & ⋯ & ⋯ & 0 & 0 \\ 0 & ... & & & O & & ⋮ \\ & & ... & & & ⋮ & ⋮ \\ ⋮ & & & ... & & & ⋮ \\ & O & & & ... & 0 & ⋮ \\ 0 & & ⋯ & & 0 & 1 & 0 \\ y & ⋯ & y^{k−1} & 0 & ⋯ & 0 & ∂f_{m}/∂y \end{matrix} ) ( g_{1}(x, y) ; ⋮ ; g_{m−1}(x, y) ; α(x, y) )

= ( g_{1}(x, y) ; ⋮ ; g_{m−1}(x, y) ; ∑_{i=1}^{k−1} y^{i}g_{i}(x, y) + α(x, y)∂f_{m}/∂y ) = ( g_{1}(x, y) ; ⋮ ; g_{m−1}(x, y) ; g_{m}(x, y) ) .

Hence the linear equation Jf(x, y)a = g(x, y) has a smooth solution

a_{1}(x, y) = g_{1}(x, y), …, a_{m−1}(x, y) = g_{m−1}(x, y), a_{m}(x, y) = α(x, y),

and, from the Lemma 2.3, M is integrable. Q.E.D.

5.1 Morin Singularities with m < n

Let f : (R^{m}, 0) → (R^{n}, 0), m < n, be a smooth stable map-germ with corank 1. Then it is right-left equivalent to the following canonical form:

f(x_{1}, …, x_{m−1}, y) = (x_{1}, …, x_{m−1}, f_{1}(x, y), …, f_{n−m+1}(x, y)),

f_{i}(x, y) = ∑_{j=1}^{k} x_{(i−1)k+j}y^{j}, 1 ≤ i < n − m + 1

f_{n−m+1}(x, y) = y^{k+1} + ∑_{j=1}^{k−1} x_{(n−m)k+j}y^{j}.

Its Jacobian matrix Jf(x, y) has the form

( \begin{matrix}1 & 0 & & & & ⋯ & ⋯ & ⋯ & & & & 0 & 0 \\ 0 & ... & & & & & & & & & O & & ⋮ \\ & & ... & & & & & & & & & ⋮ & ⋮ \\ & & ... & ... & & & & & & & & ⋮ & ⋮ \\ & & & ... & ... & & & & & & & ⋮ & ⋮ \\ ⋮ & & & & ... & ... & & & & & & ⋮ & ⋮ \\ ⋮ & & & & & ... & ... & & & & & & ⋮ \\ ⋮ & & & & & & ... & ... & & & & & ⋮ \\ & & & & & & & ... & ... & & & & ⋮ \\ & & & & & & & & ... & ... & & ⋮ & ⋮ \\ & O & & & & & & & & ... & ... & 0 & ⋮ \\ 0 & & & & ⋯ & & & & & & 0 & 1 & 0 \\ y & ⋯ & y^{k−1} & 0 & ⋯ & 0 & 0 & ⋯ & 0 & 0 & ⋯ & 0 & ∂f_{1}/∂y \\ 0 & ⋯ & 0 & y & ⋯ & y^{k−1} & 0 & ⋯ & 0 & 0 & ⋯ & 0 & ∂f_{2}/∂y \\ & ⋯ & & & & ⋯ & ⋯ & ⋯ & & & ⋯ & & ⋯ \\ 0 & & & & 0 & y & ⋯ & y^{k−1} & 0 & ⋯ & 0 & ∂f_{n−m+1}/∂y \end{matrix} ) ,

\frac{∂f_{i}}{∂y} (x, y) = x_{(i−1)k+1} + ∑_{j=2}^{k} jx_{(i−1)k+j}y^{j−1},

\frac{∂f_{n−m+1}}{∂y} (x, y) = x_{(n−m)k+1} + (k + 1)y^{k} + ∑_{j=2}^{k−1} jx_{(n−m)k+j}y^{j−1}.

Proposition 5.2 Let f : (R^{m}, 0) → (R^{n}, 0), m < n be a smooth, corank 1 stable map-germ. Let g : (R^{m}, 0) → (R^{n}, 0) be a smooth map such that

1) (f, g) : (R^{m}, 0) → T R^{n} = R^{n} × R^{n} is an embedding,

2) the image manifold-germ M = (f, g)((R^{m}, 0)) satisfies N-integrability condition. Then M is integrable.

Proof. N-integrability condition is equivalent to the fact that the linear equation 1 has a solution a(x, y) for every (x, y). By Lemma 2.3, in order to prove that M is integrable, it suffices to prove that the corresponding linear equation has a smooth solution. From the form of Jf(x, y), a_{1}(x, y), …, a_{m−1}(x, y) are uniquely determined and smooth:

a_{1}(x, y) = g_{1}(x, y), …, a_{m−1}(x, y) = g_{m−1}(x, y).

So the problem is to find a smooth solution a_{m}(x, y). Note that outside of the singular point set S(f), the solution a_{m}(x, y) is unique.

We choose a solution a(x, y) and fix it. We are going to find another solution

a_{1}(x, y) = g_{1}(x, y), …, a_{m−1}(x, y) = g_{m−1}(x, y), and α(x, y),

which is smooth.

Since a(x, y) is a solution of the equation Jf(x, y)a = g(x, y), from the form of Jacobian matrix, we see that at the points (x, y) where ∂f_{1}/∂y(x, y) = 0, we have

g_{m}(x, y) = ∑_{i=1}^{k−1} a_{i}(x, y)y^{i} = ∑_{i=1}^{k−1} g_{i}(x, y)y^{i}.

Hence we have

(g_{m}(x, y) − ∑_{i=1}^{k−1} g_{i}(x, y)y^{i}) |_{{∂f_{1}/∂y(x,y)=0}}≡ 0.

Since

\frac{∂f_{1}}{∂y} (x, y) = x_{1} + ∑_{j=2}^{k} ix_{j}y^{j−1}

is irreducible, there exists a smooth function α_{1}(x, y) ∈ E_{x,y} such that

g_{m}(x, y) − ∑_{i=1}^{k−1} g_{i}(x, y)y^{i} = α_{1}(x, y)\frac{∂f_{1}}{∂y} (x, y) = 0.

Thus (g_{1}(x, y), …, g_{m} − 1(x, y), α_{1}(x, y)) is a smooth solution of the sub equation consisting of the first m lines of the equation Jf(x, y)a = g(x, y) :

( \begin{matrix}1 & 0 & ⋯ & ⋯ & ⋯ & 0 & 0 \\ 0 & ... & & & O & & ⋮ \\ & & ... & & & ⋮ & ⋮ \\ ⋮ & & & ... & & & ⋮ \\ & O & & & ... & 0 & ⋮ \\ 0 & & ⋯ & & 0 & 1 & 0 \\ y & ⋯ & y^{k−1} & 0 & ⋯ & 0 & ∂f_{1}/∂y \end{matrix} ) ( g_{1}(x, y) ; ⋮ ; g_{m−1}(x, y) ; α(x, y) ) = ( g_{1}(x, y) ; ⋮ ; g_{m−1}(x, y) ; g_{m}(x, y) ) ,

The solution a(x, y) of the equation Jf(x, y)a(x, y) = g(x, y) must be also a solution of the sub equation

( \begin{matrix}1 & 0 & ⋯ & ⋯ & ⋯ & 0 & 0 \\ 0 & ... & & & O & & ⋮ \\ & & ... & & & ⋮ & ⋮ \\ ⋮ & & & ... & & & ⋮ \\ & O & & & ... & 0 & ⋮ \\ 0 & & ⋯ & & 0 & 1 & 0 \\ y & ⋯ & y^{k−1} & 0 & ⋯ & 0 & ∂f_{1}/∂y \end{matrix} ) ( a_{1} ; ⋮ ; a_{m−1} ; a_{m} ) = ( g_{1}(x, y) ; ⋮ ; g_{m−1}(x, y) ; g_{m}(x, y) ) ,

which is unique in the region {∂f_{1}/∂y(x, y) ≠ 0}. Thus we have

a_{m}(x, y) = α_{1}(x, y) on {∂f_{1}/∂y(x, y) ≠ 0}.

In the same way, for i = 2, …, n − m + 1 , the sub equation consisting of the first m − 1 equations and the (m + i − 1)-th equation of the equation Jf(x, y)a = g(x, y)

( \begin{matrix}1 & 0 & ⋯ & ⋯ & ⋯ & 0 & 0 \\ 0 & ... & & & O & & ⋮ \\ & & ... & & & ⋮ & ⋮ \\ ⋮ & & & ... & & & ⋮ \\ & O & & & ... & 0 & ⋮ \\ 0 & & ⋯ & & 0 & 1 & 0 \\ y & ⋯ & y^{k−1} & 0 & ⋯ & 0 & ∂f_{i}/∂y \end{matrix} ) ( g_{1}(x, y) ; ⋮ ; g_{m−1}(x, y) ; α_{i}(x, y) ) = ( g_{1}(x, y) ; ⋮ ; g_{m−1}(x, y) ; g_{m+i−1}(x, y) ) ,

has a smooth solution α_{i}(x, y) which coincides with a_{m}(x, y) in the region {∂f_{i}/∂y(x, y) ≠ 0}:

α_{i}(x, y) = a_{m}(x, y) ∀(x, y) ∈ {∂f_{i}/∂y(x, y) ≠ 0}

Thus we have

α_{1}(x, y) = α_{2}(x, y) = ⋯ = α_{n−m+1}(x, y) = a_{m}(x, y)

at every point (x, y) ∈ {∂f_{1}/∂y(x, y) ≠ 0, …, ∂f_{n−m+1}/∂y(x, y) ≠ 0}. Thus α_{1}(x, y), α_{2}(x, y), ⋯, α_{n−m+1}(x, y) coincide with each other on the open and dense set {∂f_{1}/∂y(x, y) ≠ 0, …, ∂f_{n−m+1}/∂y(x, y) ≠ 0}.

Since they are all smooth, they must coincide with each other on the whole region where they are defined. Set

α(x, y) = α_{1}(x, y) = α_{2}(x, y) = ⋯ = α_{n−m+1}(x, y).

Thus with this α(x, y), we have found a smooth solution g_{1}(x, y), …, g_{m−1}(x, y), α(x, y) of the original equation Jf(x, y)a = g(x, y). This completes the proof of the proposition. Q.E.D.

5.2 Morin Singularities with m > n

Let f : (R^{m}, 0) → (R^{n}, 0), m > n, be a stable smooth map-germ with corank 1. Then it is right-left equivalent to the following canonical form:

f(x_{1}, …, x_{n−1}, y_{1}, …, y_{m−n}, z)) = (x_{1}, …, x_{n−1}, f_{n}(x, y, z)).

f_{n}(x, y, z)) = z^{k−1} ∑_{i=1}^{k−1} x_{i}z^{i} + ∑_{i=1}^{m−n} ±y_{i}^{2}.

Proposition 5.3 Let f : (R^{m}, 0) → (R^{n}, 0) be a smooth corank 1 stable map-germ. Let g : (R^{m}, 0) → (R^{n}, 0) be a smooth map such that

1) (f, g) : (R^{m}, 0) → T R^{n} = R^{n} × R^{n} is an embedding,

2) the image manifold-germ M = (f, g)((R^{m}, 0)) satisfies N-integrability condition. Then M is integrable.

Proof. The proof of this Proposition goes the same lines as before. We may leave it as a good exercise. Q.E.D.

6 Examples and counter examples

Let us define the following four spaces of surface-germs M = (f, g)(R^{m}) ⊂ T R^{n} involved in our integrability conditions.

M_{0} = {M | M satisfies the N-integrability condition }
M = {M | M integrable }
M_{1} = {M | M = (f, g)(R^{m}), the linear equation

Jf(u)b(u) = g(u) has a smooth solution }

M_{2} = {M | M = is generic in the sense of J. Mather }

Proposition 6.1 There is a sequence of proper inclusions.

M_{2} ⊈ M_{1} ⊈ M ⊈ M_{0}.

As a prove of this proposition, at each inclusion ⊈ we give a simple example which belongs to the latter set but not to the former.

Example 6.1 An example that fulfills the N-integrability condition but is not integrable.

Let

f, g : R^{2} → R^{2}, f(u, v) = (u, u^{2}v + v^{3}), g(u, v) = (1 + v, u)
M = (f, g)(R^{2}).

Then M satisfies N-Integrability condition

g(u, v) ∈ Jf(u, v)(R^{2}),

but M is not integrable at (1, 0) = (f, g)(0, 0) ∈ M.

First (f, g) : R^{2} → T R^{2} = R^{2} × R^{2} is an embedding. Since

Jf(u, v) = ( \begin{matrix}1 & 0 \\ 2uv & u^{2} + 3v^{2} \end{matrix} ) ,

at any point (u, v) ≠ (0, 0), Jf(u, v) is a regular matrix and g(u, v) ∈ Jf(u, v)(R^{2}). At (u, v) = (0, 0),

g(0, 0) = ( \begin{matrix}1 \\ 0 \end{matrix} ) ∈ ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) R^{2}.

Thus M = (f, g)(R^{2}) satisfies N-Integrability condition.

Now let us see that M is not integrable at (1, 0) = (f, g)(0, 0) ∈ M. Suppose that (1, 0) = (f, g)(0, 0) ∈ M is an integrable point of M. Then there exists an integral curve γ : (−ϵ, ϵ) → R^{2} such that

(γ(0), γ′(0)) = (1, 0), (γ(t), γ′(t)) ∈ M, ∀t ∈ (−ϵ, ϵ).

Since (f, g) : R^{2} → M ⊂ T R^{2} = R^{2} × R^{2} is a diffeomorphism, we can consider the curve

α : (−ϵ, ϵ) → R^{2} defined by α(t) = (f, g)^{−1}(γ(t), γ′(t)).

Setting

α(t) = (u(t), v(t)), where α(0) = (u(0), v(0)) = (0, 0)

we have the equality

g(α(t)) = g(u(t), v(t)) = ( \begin{matrix}1 + v(t) \\ u(t) \end{matrix} ) = Jf(α(t))α′(t)

= ( \begin{matrix}1 & 0 \\ 2u(t)v(t) & u(t)^{2} + 3v(t)^{2} \end{matrix} ) ( \begin{matrix}u′(t) \\ v′(t) \end{matrix} ) .

Therefore we have

1 + v(t) = u′(t)  (3)
u(t) = 2u(t)v(t)u′(t) + (u(t)^{2} + 3v(t)^{2})v′(t).  (4)

Now we compare the orders of the both sides of the second equation with respect to the variable t. Since u(0) = v(0) = 0 and from the first equation, we see that

the order of the left hand side u(t) is equal to 1.

On the other hand, since u(0) = v(0) = 0, we see that

the order of the right hand side 2u(t)v(t)u′(t) + (u(t)^{2} + 3v(t)^{2})v′(t) ≥ 2.

This is a contradiction. Thus we see that there is no integral curve γ : (−ϵ, ϵ) → R^{2} such that

(γ(0), γ′(0)) = (1, 0), (γ(t), γ′(t)) ∈ M, ∀t ∈ (−ϵ, ϵ),

and (1, 0) = (f, g)(0, 0) ∈ M is not an integrable point of M.

Example 6.2 An example that has no smooth solution of the Jacobian linear equation but Integrable

For n > 2, set

f, g : R^{2} → R^{2}, f(u, v) = (u, v^{n}), g(u, v) = (1, v),
M = (f, g)(R^{2}).

Then M is an integrable manifold, although the Jacobian linear equation g(u, v) = Jf(u, v)b(u) has no smooth solution.

First we easily see that (f, g) : R^{2} → T R^{2} = R^{2} × R^{2} is an embedding. Next let us check the N-Integrability condition. Since

Jf(u, v) = ( \begin{matrix}1 & 0 \\ 0 & nv^{n−1} \end{matrix} ) ,

at any point (u, v) with v ≠ 0, Jf(u, v) is a regular matrix and g(u, v) ∈ Jf(u, v)(R^{2}). At (u, 0),

g(u, 0) = ( \begin{matrix}1 \\ 0 \end{matrix} ) ∈ ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) R^{2}.

Thus M = (f, g)(R^{2}) satisfies N-Integrability condition.

Now consider the Jacobian linear equation g(u, v) = Jf(u, v)b(u):

g(u, v) = ( \begin{matrix}1 \\ v \end{matrix} ) = ( \begin{matrix}1 & 0 \\ 0 & nv^{n−1} \end{matrix} ) ( \begin{matrix}b_{1}(u, v) \\ b_{2}(u, v) \end{matrix} ) .

If there exist a smooth solution b_{1}(u, v), b_{2}(u, v), it must satisfy

v = nv^{n−1}b_{2}(u, v),

which is impossible for a smooth function b_{2}(u, v) and for n > 2. Thus the Jacobian linear equation g(u, v) = Jf(u, v)b(u, v) has no smooth solutions.

Now let us check that M is integrable. The points (f, g)(u, v) ∈ M where v ≠ 0 are integrable. For around the points (u, v) where v ≠ 0, the Jacobian linear equation g(u, v) = Jf(u, v)b(u, v) has smooth solutions.

Now let check the points (f, g)(u, 0) = (u, 0, 1, 0) ∈ M. Consider the submanifold

S = { (u, v) ∈ R^{2} | v = 0 } ⊂ R^{2}

and restrict the Jacobian linear equation to S:

g(u, 0) = ( \begin{matrix}1 \\ 0 \end{matrix} ) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ( \begin{matrix}b_{1}(u, 0) \\ b_{2}(u, 0) \end{matrix} )

Then we have a smooth solution on S:

b_{1}(u, 0) = 1, b_{2}(u, 0) = 0.

The solution (b_{1}(u, 0), b_{2}(u, 0)) = (1, 0) is tangent to the submanifold S at every point (u, 0) ∈ S. Consider the integral curves

α(t) = (u + t, 0) of the vector field \frac{∂}{∂u} on S

and the curves

γ : R → R^{2}, γ(t) = f(α(t)) = f(u + t, 0) = (u, 0).

Then we have

(γ(t), γ′(t)) = (u + t, 1) = (f, g)(u + t, 0) ∈ (f, g)(S) ⊂ M,
(γ(0), γ′(0)) = (u, 1) = (f, g)(u, 0).

Thus the points of the form (γ(0), γ′(0)) = (u, 1) = (f, g)(u, 0). are also integrable and hence M is integrable.

Example 6.3 Non-generic but there exists a smooth solution of the Jacobian linear equation.

Let

f, g : R^{2} → R^{2}, f(u, v) = (u, f_{2}(u, v)), g(u, v) = (1 + v, (1 + v)∂f_{2}/∂u(u, v)),

such that \frac{∂^{2}f_{2}}{∂u∂v} (0, 0) = \frac{∂^{2}f_{2}}{∂v^{2}} (0, 0) = 0,
M = (f, g)(R^{2}).

Then Jf : R^{2} → E(2, 2) is not transversal to S_{1} but the Jacobian linear equation

g(u, v) = Jf(u, v)b(u)

has a smooth solution, where E(n, m) is the space of n × m real matrices and S_{r} is the submanifold of E(n, m) consisting of matrices of rank r.

First we easily see that (f, g) : R^{2} → T R^{2} = R^{2} × R^{2} is an embedding.

Next let us check that the Jacobian linear equation

g(u, v) = Jf(u, v)b(u)

has a smooth solution. Since

Jf(u, v) = ( \begin{matrix}1 & 0 \\ ∂f_{2}/∂u(u, v) & ∂f_{2}/∂v(u, v) \end{matrix} ) ,

we see that

g(u, v) = ( \begin{matrix}1 + v \\ (1 + v)∂f_{2}/∂u(u, v) \end{matrix} ) = ( \begin{matrix}1 & 0 \\ ∂f_{2}/∂u(u, v) & ∂f_{2}/∂v(u, v) \end{matrix} ) ( \begin{matrix}1 + v \\ 0 \end{matrix} ) .

Thus the Jacobian linear equation

g(u, v) = Jf(u, v)b(u)

has the smooth solution (1 + v, 0).

On the other hand, since

\frac{∂^{2}f_{2}}{∂u∂v} (0, 0) = \frac{∂^{2}f_{2}}{∂v^{2}} (0, 0) = 0,

then Jf : R^{2} → E(2, 2) is not transversal to S_{1} at (0, 0). Because Jf : R^{2} → E(2, 2) is transversal to S_{1} at (0, 0) if and only if ∂detJf/∂u(0, 0) ≠ 0 or ∂detJf/∂v(0, 0) ≠ 0.

7 Smooth solutions of linear equations and irreducibility of determinants

In Section 4, by using J. Mather’s result on smooth solutions of linear equations, we gave a generic sufficient condition for a submanifold M of T R^{n} = R^{n}×R^{n} to be integrable. Now we give an extension of J. Mather’s theorem weakening his transversality condition to a kind of irreducibility of the determinant of coefficient matrices of the linear equations.

In order to make our argument simpler, we consider real analytic category rather than C^{∞} category.

Since our problem of integrability of submanifolds of T R^{n} = R^{n} × R^{n} is a local problem, we need only the local version of J. Mather’s theorem, which we recall here. Consider a n × m matrix A(x) of real analytic function germs defined at the origin of R^{k} :

A(x) = ( \begin{matrix}a_{11}(x) & & a_{12}(x) & & & … & & a_{1m}(x) \\ a_{21}(x) & & a_{22}(x) & & & … & & a_{2m}(x) \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ a_{n1}(x) & & a_{n2}(x) & & & … & & a_{nm}(x) \end{matrix} ) ,

where a_{ij} : (R^{k}, 0) → R are germs at 0 ∈ R^{k} of real analytic functions defined on R^{k}, and a column vector

g(x) = ( g_{1}(x) ; ⋮ ; g_{n}(x) ) ,

where g_{i} : (R^{k}, 0) → R are also germs at 0 ∈ R^{k} of real analytic functions defined on R^{k}. Consider the equation A(x)b = g(x), where b is a column vector of length m.

Let E(n, m) denote the space of n × m matrices of real numbers. Let S_{r} denote the subset of E(n, m) consisting of matrices of rank r. Then S_{r} is a submanifold of codimension (n − r) × (m − r) in E(n, m).

Definition 7.1 An n×m matrix A(x) of real analytic function germs, A : (R^{k}, 0) → E(n, m), is said to be generic if A : (R^{k}, 0) → E(n, m) is transversal to S_{r}, where r = rankA(0).

J. Mather’s Theorem If A(x) is generic and the equation A(x)b = g(x) has a solution for every x ∈ R^{k} sufficiently close to 0 ∈ R^{k}, then this equation has a real analytic solution germ b(x).

7.1 Extension of J. Mather’s theorem: Case where m = n

Let A(x) be m × m matrix whose entries a_{ij}(x) are germs at 0 ∈ R^{k} of real analytic functions defined on R^{k}. Let g : (R^{k}, 0) → R^{m=n} be a real analytic map germ such that g(x) ∈ A(x)R^{m}, ∀x, where we regard g(x) as a column vector of real analytic function germs. Consider the linear equation

A(x)b = g(x).  (5)

Let O_{k} denote the ring of germs at 0 ∈ R^{k} of real analytic functions of k variables.

Proposition 7.1 If the ideal < detA(x) > in O_{k} generated by the determinant of the matrix A(x) has property of zeroes, (i.e. if any function h(x) vanish on the variety defined by this ideal then h(x) belongs to the ideal), then the linear equation (5) A(x)b = g(x) has a real analytic solution b(x).

Proof. (cf. J. Mather [12], p.190) Let Ã(x) denote the cofactor matrix of A(x). Then we have

Ã(x)A(x) = A(x)Ã(x) = detA(x)I_{m},

where I_{m} is the identity matrix of size m. Then the equation (5) is equivalent to

detA(x)b = Ã(x)g(x).

Therefore the component functions g̃_{1}(x), …, g̃_{m}(x) of column vector Ã(x)g(x) vanish on the variety {detA(x) = 0}. The hypothesis of Proposition 7.1 implies that detA(x) divides the component functions g̃_{1}(x), …, g̃_{m}(x) and

b(x) = ( b_{1}(x) ; ⋮ ; b_{m}(x) ) = \frac{1}{detA(x)}Ã(x)g(x) = ( g̃_{1}(x)/detA(x) ; ⋮ ; g̃_{m}(x)/detA(x) ) ,

is a real analytic solution of the equation (5). Q.E.D.

Now we investigate conditions for the ideal < detA(x) > in O_{k} to have property of zeroes.

Let

O_{k,C}

denote the ring of germs at 0 ∈ C^{k} of complex analytic functions of k variables. Any real analytic function germ f(x) ∈ O_{k} can be uniquely extended to a complex analytic function germ f̃(z) ∈ O_{k,C}, which we call the complexification of f(x). Identifying f(x) ∈ O_{k} with its complexification f̃(z) ∈ O_{k,C}, we may regard O_{m} as a subring of O_{k,C}.

An element a of a ring R is said to be irreducible (in R) if a is not a unit of R and satisfies the condition

a = bc ⟹ b or c is a unit of R.

Proposition 7.2 If a real analytic function germ f(x) ∈ O_{k} is irreducible in O_{k} as well as so is its complexification f̃(z) in O_{k,C}, then the ideal < f(x) > in O_{k} generated by f(x) has property of zeros.

Proof. Let g(x) be an element of O_{k} vanishing on the germ at 0 ∈ R^{k} of the variety

V (f) = {x ∈ R^{k} | f(x) = 0}.

Then its complexification g̃(z) also vanishes on the germ at 0 ∈ R^{k} of complex variety

Ṽ (f̃) = {z ∈ C^{k} | f̃(x) = 0}.

( See, for example, [15], Chapter V, Proposition 1 (p.91)).

Now from Hilbert’s Nullstellensatz, for some positive power n, we have

g̃^{n}(z) ∈< f̃(z) > .

Since f̃(z) is irreducible in O_{k,C}, f̃(z) divides g̃(z) in O_{k,C} :

g̃(z) = b̃(z)f̃(z), ∃b̃(z) ∈ O_{k,C}.

Consider g̃(z), b̃(z), f̃(z) as power series. Then , since the coefficients of g̃(z),and f̃(z) are real, the coefficients of b̃(z) are also real. Thus b̃(z) is the complexification of a real analytic function b(x). Thus we have g(x) = b(x)f(x), which implies f(x) divides g(x) in O_{k}. Q.E.D.

Proposition 7.3 Let f(x) ∈ O_{k} be a real analytic function germ. If f(x) has a decomposition

f(x) = f_{1}(x)f_{2}(x) ⋯ f_{ℓ}(x), f_{i}(x) ∈ O_{k}

such that

1) f_{1}(x), …, f_{ℓ}(x) are all irreducible in O_{k} and that any factor f_{i}(x) is not divisible by other factors f_{j}(x), i ≠ j, and such that

2) the complexifications f̃_{1}(z), f̃_{2}(z), ⋯, f̃_{ℓ}(z) of f_{1}(x), f_{2}(x), ⋯, f_{ℓ}(x) are all irreducible in O_{k,C}, then the ideal < f(x) > in O_{k} generated by f(x) has property of zeros.

Proof. Let g(x) be an element of O_{k} vanishing on the germ at 0 ∈ R^{k} of the variety

V (f) = {x ∈ R^{k} | f(x) = 0}.

Then g(x) vanishes on the germ at 0 ∈ R^{k} of the variety

V (f_{i}) = {x ∈ R^{k} | f_{i}(x) = 0} ( ⊂ V (f) ).

Now f_{i}(x) satisfies the condition in Proposition 7.2, f_{i}(x) divides g(x). Since f_{i}(x)’s are all irreducible and any factor f_{i}(x) is not divisible by other factorsf_{j}(x), i ≠ j, then f(x) = f_{1}(x)f_{2}(x) ⋯ f_{ℓ}(x) divides g(x). Q.E.D.

Let A(x) be m × m matrix whose entries a_{ij}(x) are germs at 0 ∈ R^{k} of real analytic functions defined on R^{k}. Let g : (R^{k}, 0) → R^{m} be a real analytic map germ such that g(x) ∈ A(x)R^{m}, ∀x. Consider here again the linear equation (5)

A(x)b = g(x).

Combining Proposition 7.1 with Propositions 7.2 and 7.3, we obtain

Proposition 7.4 If detA(x) satisfies one of the following two conditions I and II, then the linear equation (5) has a real analytic solution b(x).

Condition I. detA(x) is irreducible in O_{k} as well as so is its complexificationin detÃ(z) in O_{k,C}, Condition II detA(x) has a decomposition

detA(x) = α_{1}(x)α_{2}(x) ⋯ α_{ℓ}(x), α_{i}(x) ∈ O_{k}

such that

1) α_{1}(x), …, α_{ℓ}(x) are all irreducible in O_{k} and that any factor α_{i}(x) is not divisible by other factors α_{j}(x), i ≠ j, and such that

2) the complexifications α̃_{1}(z), α̃_{2}(z), ⋯, α̃_{ℓ}(z) of α_{1}(x), α_{2}(x), ⋯, α_{ℓ}(x) are all irreducible in O_{k,C}.

Now let’s go back further to the original problem of integrability. The following is an immediate consequence of Proposition 7.4.

Theorem 7.1 Let f : R^{m} → R^{m} be a real analytic map and g : R^{m} → R^{n} be a real analytic map tangent to f. If detJf(x) satisfies one of the conditions I and II in Proposition 7.4, then the germ at (f(0), g(0)) of M = (f, g)(R^{m}) ⊂ T R^{m} is integrable.

7.2 Extension of J. Mather’s theorem: Case where m < n

In this section we observe the case where m < n. Suppose m < n. We consider an n × m matrix A(x) of real analytic function germs defined at the origin of R^{k} :

A(x) = ( \begin{matrix}a_{11}(x) & & a_{12}(x) & & & … & & a_{1m}(x) \\ a_{21}(x) & & a_{22}(x) & & & … & & a_{2m}(x) \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ a_{n1}(x) & & a_{n2}(x) & & & … & & a_{nm}(x) \end{matrix} ) ,

where a_{ij} : (R^{k}, 0) → R are germs at 0 ∈ R^{k} of real analytic functions defined on R^{k}, and a column vector

g(x) = ( g_{1}(x) ; ⋮ ; g_{n}(x) ) ,

where g_{i} : (R^{k}, 0) → R are also germs at 0 ∈ R^{k} of real analytic functions defined on R^{k}. Consider the equation

A(x)b = g(x),  (6)

where b is a column vector of length m.

Suppose that the rank of the matrix A(0) at the origin 0 ∈ R^{k} is r. Then we may suppose that the matrix A(x) be of the form

A(x) = ( \begin{matrix}I_{r} & O \\ O & A′ \end{matrix} ) , where I_{r} is an identity matrix of size r

and A′ = ( \begin{matrix}a_{r+1,r+1}(x) & & & a_{r+1,r+2}(x) & & ⋯ & & a_{r+1,m}(x) \\ a_{r+2,r+1}(x) & & & a_{r+2,r+2}(x) & & ⋯ & & a_{r+2,m}(x) \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ a_{n,r+1}(x) & & & a_{n,r+2}(x) & & ⋯ & & a_{n,m}(x) \end{matrix} )

Consider the set

I = {I = (i_{1}, i_{2}, …, i_{m−r}) | i_{j} ∈ N, r < i_{1} < i_{2} < … < i_{m−r} ≤ n},

and for each element I = (i_{1}, i_{2}, …, i_{m−r}) ∈ I consider the (m−r)×(m−r) minor of A′(x):

det_{I}A(x) = det ( \begin{matrix}a_{i_{1},r+1}(x) & & a_{i_{1},r+2}(x) & & ⋯ & & a_{i_{1},m}(x) \\ a_{i_{2},r+1}(x) & & a_{i_{2},r+2}(x) & & ⋯ & & a_{i_{2},m}(x) \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ a_{i_{m−r},r+1}(x) & & a_{i_{m−r},r+2}(x) & & ⋯ & & a_{i_{m−r},m}(x) \end{matrix} ) .

Proposition 7.5 If the equation A(x)b = g(x) has a solution b(x)for every x ∈ R^{k} and if for every element I ∈ I the ideal < det_{I}A(x) > in O_{k} generated by the (m − r) × (m − r) minor det_{I}A(x) of the matrix A(x) has property of zeroes, then the linear equation A(x)b = g(x) has a real analytic solution b(x).

Proof. Although the proof of this proposition is almost the same as the argument in §7( pp.191-192) of J. Mather [12], we recall it here.

First we investigate the equation (6) at points x ∈ R^{k} where rank A(x) = m = min(m, n). Set

Ω = {x ∈ R^{k} | rank A(x) = m},
Σ = R^{k} − Ω = {x ∈ R^{k} | rank A(x) < m}.

Lemma 7.1 1) Ω is an open dense set of R^{k}.

2) For every point x ∈ Ω, the equation (6) has a unique solution, which we denote by b_{Ω}(x).

3) The unique solution b_{Ω}(x), x ∈ Ω, is analytic in Ω.

Proof. 1) Is trivial.

2) Since rank A(x) = m = min(m, n), if there exists a solution b(x), which does from the assumption of the Lemma, then b(x) is a unique solution of (6).

3) First note that equation (1) is of the form

( g_{1}(x) ; ⋮ ; g_{n}(x) ) = ( \begin{matrix}I_{r} & O \\ O & A′ \end{matrix} ) ( b_{1}(x) ; ⋮ ; b_{m}(x) ) .

Since x ∈ Ω, there exists I = (i_{1}, i_{2}, …, i_{m−r}) ∈ I such that the sub matrix A′_{I}(x) has rank m − r, where

A′_{I} = ( \begin{matrix}a_{i_{1},r+1}(x) & & a_{i_{1},r+2}(x) & & ⋯ & & a_{i_{1},m}(x) \\ a_{i_{2},r+1}(x) & & a_{i_{2},r+2}(x) & & ⋯ & & a_{i_{2},m}(x) \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ a_{i_{m−r},r+1}(x) & & a_{i_{m−r},r+2}(x) & & ⋯ & & a_{i_{m−r},m}(x) \end{matrix} ) .

Set

A_{I}(x) = ( \begin{matrix}I_{r} & O \\ O & A′_{I} \end{matrix} ) ,

and consider the sub equation

g_{I}(x) = A_{I}(x)b,
where g_{I}(x) = ^{t}(g_{1}(x), …, g_{r}(x), g_{i_{1}}(x), …, g_{i_{m−r}}(x)).

Since b_{Ω}(x) is a solution of (6), b_{Ω}(x) is also a solution of the sub equation g_{I}(x) = A_{I}(x)b, which is analytic since g_{I}(x) is analytic and detA_{I}(x) ≠ 0. Q.E.D.

Now consider the solution at points x ∈ Σ where rankA(x) < m. Take any I = (i_{1}, i_{2}, …, i_{m−r}) ∈ I and consider sub equation

g_{I}(x) = A_{I}(x)b,

where A_{I}(x) and g_{I}(x) are defined in the same way as those in the proof of Lemma 7.1. Then, from the assumption of Proposition 7.5, the ideal < detA_{I}(x) > satisfies the condition posed in Proposition 7.1 that it has property of zeros. Therefore the sub equation g_{I}(x) = A_{I}(x)b has an analytic solution b_{I}(x).

Now consider the set

Ω_{I} = {x ∈ R^{k} | det A_{I}(x) ≠ 0} ⊂ Ω.

Note that Ω_{I} is also open and dense in R^{k}.

We compare the restriction (b_{I} |_{Ω_{I})} (x) of b_{I}(x) to Ω_{I} with the restriction (b_{Ω} |_{Ω_{I}})(x) of b_{Ω}(x) to Ω_{I}. Since both of (b_{I} |_{Ω_{I}})(x) and (b_{Ω} |_{Ω_{I}})(x) are solutions of the sub equation

g_{I}(x) = A_{I}(x)b, x ∈ Ω_{I},

and the solution of the sub equation g_{I}(x) = A_{I}(x)b is unique on the region Ω_{I} ⊂ Ω, we have

b_{I}(x) = b_{Ω}(x) ∀x ∈ Ω_{I}.

Now take any other J ∈ I, J ≠ I and consider the sub equation

g_{J}(x) = A_{J}(x)b.

With the same reason, the sub equation g_{J}(x) = A_{J}(x)b has an analytic solution b_{J}(x) and we have

b_{J}(x) = b_{Ω}(x) ∀x ∈ Ω_{J}.

Hence we have

b_{J}(x) = b_{I}(x) = b_{Ω}(x) ∀x ∈ Ω_{J} ∩ Ω_{I}.

Thus b_{J}(x) and b_{I}(x) coincide on an open dense subset Ω_{J} ∩ Ω_{I} of R^{k} and they are analytic. Therefore they coincide on the whole R^{k}. Set

b(x) = b_{I}(x) ∀I ∈ I.

Then b(x) is an analytic solution of all the sub equations g_{I}(x) = A_{I}(x)b, and it is an analytic solution of the whole equation g(x) = A(x)b. Thus we proved the Proposition 7.5 Q.E.D.

Using Proposition 7.5 and Propositions 7.2 and 7.3, in the same way as in the proof of Proposition 7.4, we can prove the following theorem :

Proposition 7.6 If the equation A(x)b = g(x) has a solution b(x) for every x ∈ R^{k} and if, for every element I ∈ I, det_{I}A(x) satisfies one of the two conditions I and II in Proposition 7.4, then the linear equation (1) has a real analytic solution b(x).

Now let’s go back to the original problem of integrability. The following is an immediate consequence of Proposition 7.6.

Theorem 7.2 Let f : R^{m} → R^{n}, m < n, be a real analytic map and g : R^{m} → R^{n} be a real analytic map tangent to f. If for every element I ∈ I, det_{I}Jf(x) satisfies one of the two conditions I and II in Proposition 7.4, then the germ at (f(0), g(0)) of M = (f, g)(R^{m}) ⊂ T R^{m} is integrable.

8 Integrability of implicit Hamiltonian systems

Now we consider the special class of implicit differential systems - the generalized Hamiltonian systems, which play an important role in field theory and mechanics.

Let (R, ω) be the symplectic Euclidean space, endowed with the symplectic structure in the Darboux form, ω = ∑_{i=1}^{n} dy_{i} ∧ dx_{i}. By the vector bundle morphism β : T R^{2n} ∋ u → u⌋ω ∈ T^{∗}R^{2n} we introduce the canonical symplectic structure ω̄ on T R^{2n}, namely the pullback of the Liouville symplectic form dθ defined on T^{∗}R^{2n}.

ω̄ = β^{∗}dθ = ∑_{i=1}^{n}(dẏ_{i} ∧ dx_{i} − dẋ_{i} ∧ dy_{i}),

where (p, q) = ((x, y), (ẋ, ẏ)) are coordinates on T R^{2n} ≡ R^{2n} × R^{2n}

Definition 8.1 A smooth submanifold M ⊂ T R^{2n} ≡ R^{2n} × R^{2n} is called generalized Hamiltonian system if M is isotropic according to ω̄, i.e. ω̄ |_{M}= 0. In this case dimM ≤ 2n. The case of M such that dimM = 2n is called the implicit Hamiltonian system.

By the Hörmander-Arnold-Weinstein generating family representation of M [1], we have the following Lemma.

Lemma 8.1 Let M ⊂ T R^{2n} ≡ R^{2n} × R^{2n} be an implicit Hamiltonian system, and let (p, q) ∈ M. Suppose that corankd(π_{1} |_{M})(p, q) = k. Then there exists an open neighbourhood O of (p, q) in T R^{2n} and a smooth function F : R^{2n} × R^{k} → R defined on some open neighbourhood of (p, 0), such that

M ∩ O = \{((x, y), (ẋ, ẏ)) ∈ O; ∃_{λ∈R^{k}} such that ẋ_{i} = \frac{∂F}{∂y_{i}} (x, y, λ),  (7)
ẏ_{j} = −\frac{∂F}{∂x_{j}} (x, y, λ), 0 = \frac{∂F}{∂λ_{l}} (x, y, λ)\},  (8)

where 1 ≤ i, j ≤ n, 1 ≤ l ≤ k, and such that

rank(\frac{∂^{2}F}{∂x_{j}∂λ_{l}} , \frac{∂^{2}F}{∂y_{i}∂λ_{l}} , \frac{∂^{2}F}{∂λ_{s}∂λ_{l}})(p, 0) = k, and \frac{∂^{2}F}{∂λ_{s}∂λ_{r}} (p, 0) = 0,  (9)

where 1 ≤ s, r ≤ k,

In what follows we will consider the Hamiltonian case, i.e. dimM = 2n, and because we are considering the local properties of M, we assume that M is generated by some generating family-germ F as in the above Lemma 8.1 and denote it also by M_{F}.

Lemma 8.2 Let M_{F} ⊂ T R^{2n} ≡ R^{2n} × R^{2n} be an implicit Hamiltonian system generated by the Morse family germ F : (R^{2n} × R^{k}, (0, 0)) → R, then the N-integrability condition of M_{F} is equivalent to the existence of the solution μ = (μ_{1}, …, μ_{n}) ∈ R^{k} of the following equation

( \begin{matrix}\frac{∂^{2}F}{∂λ_{1}∂λ_{1}} & … & \frac{∂^{2}F}{∂λ_{1}∂λ_{k}} \\ ⋮ & ⋮ & ⋮ \\ \frac{∂^{2}F}{∂λ_{k}∂λ_{1}} & … & \frac{∂^{2}F}{∂λ_{k}∂λ_{k}} \end{matrix} ) ( μ_{1} ; ⋮ ; μ_{k} ) = ( \{\frac{∂F}{∂λ_{1}} , F \} ; ⋮ ; \{\frac{∂F}{∂λ_{k}} , F \} ) ,  (10)

for each (p, q) ∈ M, where {., .} denotes the Poisson bracket on R^{2n} induced by ω.

Proof. Assume that γ : (−ε, ε) → R^{2n} and δ : (−ε, ε) → R^{k}, γ(0) = (x, y), δ(0) = λ, is a smooth solution of (7), throug the point (x, y, λ) of M. Then we have

\frac{∂F}{∂λ_{l}} (γ(t), δ(t)) ≡ 0, ∀_{t∈(−ε,ε)}, 1 ≤ l ≤ k,

and also the derivative.

∑_{i=1}^{k} \frac{∂^{2}F}{∂λ_{i}∂λ_{l}} (γ(t), δ(t))δ′_{i}(t) + ∑_{i=1}^{k} \frac{∂^{2}F}{∂x_{i}∂λ_{l}} (γ(t), δ(t))ẋ_{i} + ∑_{i=1}^{k} \frac{∂^{2}F}{∂y_{i}∂λ_{l}} (γ(t), δ(t))ẏ_{i} =

∑_{i=1}^{k} \frac{∂^{2}F}{∂λ_{i}∂λ_{l}} (γ(t), δ(t))δ′_{i}(t) + ∑_{i=1}^{k} \frac{∂^{2}F}{∂x_{i}∂λ_{l}} (γ(t), δ(t))\frac{∂F}{∂y_{i}} (γ(t), δ(t))

− ∑_{i=1}^{k} \frac{∂^{2}F}{∂y_{i}∂λ_{l}} (γ(t), δ(t))\frac{∂F}{∂x_{i}} (γ(t), δ(t)), 1 ≤ l ≤ k.

Taking γ(0) = (x, y), δ(0) = λ, we get the solvability of the equation

∑_{i=1}^{k} \frac{∂^{2}F}{∂λ_{i}∂λ_{l}} (x, y, λ)μ_{i} − \{\frac{∂F}{∂λ_{l}} , F \} = 0, 1 ≤ l ≤ k

on the smooth manifold given by equations, \frac{∂F}{∂λ_{l}} (x, y, λ) = 0. Q.E.D.

N-integrability condition is the necessary condition for integrability of M. Now we give an example which fulfills the N-integrability condition but it is not integrable.

Example 8.1 As in the case of Example 6.1 in Section 6, we show that the N-integrability condition is also only a necessary condition for implicit Hamiltonian systems.

Let us consider the implicit Hamiltonian system surface-germ M ⊂ T R^{2} ≡ R^{2} × R^{2} around (p, q) = (0, 0, 1, 0). M is generated by the generating family-germ F : R^{2}×R^{2} → R, at (x, y, λ_{1}, λ_{2}) = (0, 0, 1, 0) of the form

F (x, y, λ) = \frac{1}{2}λ_{2}^{2}(1 − (λ_{1} − 1)^{2}) − \frac{1}{4}(λ_{1} − 1)^{4} + λ_{1}y − λ_{2}x.

So the corresponding equations for M one can write in the form.

ẋ = \frac{∂F}{∂y} (x, y, λ) = λ_{1},
ẏ = −\frac{∂F}{∂x} (x, y, λ) = λ_{2},
0 = \frac{∂F}{∂λ_{1}} (x, y, λ) = y − λ_{2}^{2}(λ_{1} − 1) − (λ_{1} − 1)^{3},
0 = \frac{∂F}{∂λ_{2}} (x, y, λ) = −x + λ_{2}(1 − (λ_{1} − 1)^{2}).

Certainly N-integrability condition is fulfilled around (0, 0, 1, 0). The determinant

( \begin{matrix}\frac{∂^{2}F}{∂λ_{1}∂λ_{1}} & \frac{∂^{2}F}{∂λ_{1}∂λ_{2}} \\ \frac{∂^{2}F}{∂λ_{2}∂λ_{1}} & \frac{∂^{2}F}{∂λ_{2}∂λ_{2}} \end{matrix} ) (x, y, λ)

has isolated zero at (0, 0, 1, 0) and so there exists a solution of the linear equation

( \begin{matrix}\frac{∂^{2}F}{∂λ_{1}∂λ_{1}} & \frac{∂^{2}F}{∂λ_{1}∂λ_{2}} \\ \frac{∂^{2}F}{∂λ_{2}∂λ_{1}} & \frac{∂^{2}F}{∂λ_{2}∂λ_{2}} \end{matrix} ) ( \begin{matrix}μ_{1} \\ μ_{2} \end{matrix} ) = ( \begin{matrix}\{\frac{∂F}{∂λ_{1}} , F \} \\ \{\frac{∂F}{∂λ_{2}} , F \} \end{matrix} ) ,

in some neighbourhood outside of (0, 0, 1, 0). At that point we have

( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}μ_{1} \\ μ_{2} \end{matrix} ) = ( \begin{matrix}0 \\ 1 \end{matrix} ) ,

and it is also solvable. So N-integrability condition is fulfilled.

If M is integrable, say at (p, q) = (0, 0, 1, 0), then there exists an integral curve γ : (−ε, ε) → R^{2} such that

(γ(0), γ′(0)) = (0, 0, 1, 0) end (γ(t), γ′(t)) ∈ M∀_{t∈(−ε,ε)}.

So there exists a smooth curve λ(t) = (λ_{1}(t), λ_{2}(t)), λ(0) = (1, 0) such that

( \begin{matrix}\frac{∂^{2}F}{∂λ_{1}∂λ_{1}} & \frac{∂^{2}F}{∂λ_{1}∂λ_{2}} \\ \frac{∂^{2}F}{∂λ_{2}∂λ_{1}} & \frac{∂^{2}F}{∂λ_{2}∂λ_{2}} \end{matrix} ) (γ(t), λ(t)) ( \begin{matrix}λ′_{1} \\ λ′_{2} \end{matrix} ) = ( \begin{matrix}−λ_{2}(t) \\ λ_{1}(t)\} \end{matrix} ) ,

i.e. the following equations are fulfilled.

λ_{2}(t) = 3(λ_{1}(t) − 1)^{2}λ′_{1}(t) + λ_{2}^{2}(t)λ′_{1}(t) + 2λ_{2}(t)(λ_{1}(t) − 1)λ′_{2}(t)  (11)
λ_{1}(t) = (1 − (λ_{1}(t) − 1)^{2})λ′_{2}(t) − 2λ_{2}(t)(λ_{1}(t) − 1)λ′_{1}(t).  (12)

From the equation (12) we get

λ′_{2}(0) = 1,

so from the first equation (11) we have a contradiction: order of λ_{2}(t) at 0 is equal 1, but order of

3(λ_{1}(t) − 1)^{2}λ′_{1}(t) + λ_{2}^{2}(t)λ′_{1}(t) + 2λ_{2}(t)(λ_{1}(t) − 1)λ′_{2}(t)

at 0 is greater than equal to 2.

Let M̃_{F} ⊂ T R^{2n}×R^{k} ≡ R^{2n}×R^{2n}×R^{k} denotes a manifold-germ defined by equations (7) in Lemma 8.1. Consider the projection

π : R^{2n} × R^{2n} × R^{k} → R^{2n} × R^{k}, (p, q, λ) → (p, λ)

and the image C_{F} = π(M̃_{F}), which is a smooth manifold-germ by transversality condition (9) in Lemma 8.1.

Let

H = ( \begin{matrix}\frac{∂^{2}F}{∂λ_{1}∂λ_{1}} (x, y, λ) & … & \frac{∂^{2}F}{∂λ_{1}∂λ_{k}} (x, y, λ) \\ ⋮ & ⋮ & ⋮ \\ \frac{∂^{2}F}{∂λ_{k}∂λ_{1}} (x, y, λ) & … & \frac{∂^{2}F}{∂λ_{k}∂λ_{k}} (x, y, λ) \end{matrix} ) .

By

H̃ = ( \begin{matrix}[\frac{∂^{2}F}{∂λ_{1}∂λ_{1}} (x, y, λ)] & … & [\frac{∂^{2}F}{∂λ_{1}∂λ_{k}} (x, y, λ)] \\ ⋮ & ⋮ & ⋮ \\ [\frac{∂^{2}F}{∂λ_{k}∂λ_{1}} (x, y, λ)] & … & [\frac{∂^{2}F}{∂λ_{k}∂λ_{k}} (x, y, λ)] \end{matrix} )

we denote the k × k matrix of corresponding minors in (p, q) ∈ M_{F} . Then, by Cramer’s rule the Equation (10) can be rewritten in the form (detH)μ = H̃\{\frac{∂F}{∂λ} , F \}, i.e.

(detH)(x, y, λ)μ_{i} = ∑_{j=1}^{k}[\frac{∂^{2}F}{∂λ_{i}∂λ_{j}}]\{\frac{∂F}{∂λ_{j}} , F \}(x, y, λ).

So we have an equivalent formulation of the N-integrability condition.

Corollary 8.1 Let M_{F} ⊂ T R^{2n} ≡ R^{2n} × R^{2n} be an implicit Hamiltonian system generated by the Morse family germ F : (R^{2n} × R^{k}, (0, 0)) → R, then the N-integrability condition of M_{F} is equivalent to the following condition.

∑_{j=1}^{k}[\frac{∂^{2}F}{∂λ_{i}∂λ_{j}}]\{\frac{∂F}{∂λ_{j}} , F \} |_{\{detH=0\}∩C_{F}} ≡ 0

Now we have the result corresponding to Lemma 2.3.

Lemma 8.3 Let F : R^{2n} × R^{k} → R be smooth generating family for M_{F}. If the linear equation

( \begin{matrix}\frac{∂^{2}F}{∂λ_{1}∂λ_{1}} (x, y, λ) & … & \frac{∂^{2}F}{∂λ_{1}∂λ_{k}} (x, y, λ) \\ ⋮ & ⋮ & ⋮ \\ \frac{∂^{2}F}{∂λ_{k}∂λ_{1}} (x, y, λ) & … & \frac{∂^{2}F}{∂λ_{k}∂λ_{k}} (x, y, λ) \end{matrix} ) ( μ_{1}(x, y, λ) ; ⋮ ; μ_{k}(x, y, λ) ) = ( \{\frac{∂F}{∂λ_{1}} , F \}(x, y, λ) ; ⋮ ; \{\frac{∂F}{∂λ_{k}} , F \}(x, y, λ) ) ,

has a smooth solution (μ_{1}(x, y, λ), …, μ_{k}(x, y, λ)) on the critical manifold

C_{F} = \{(x, y, λ); \frac{∂F}{∂λ_{i}} (x, y, λ) = 0, 0 ≤ i ≤ k\}.

Then the manifold M_{F} is integrable.

Proof. We can easily see, on the basis of Lemma 8.1 that if we find the smooth solution of the linear equation (10) with respect to μ, then the system of equations

ẋ_{i} = \frac{∂F}{∂y_{i}} (x, y, λ), 1 ≤ i ≤ n
ẏ_{j} = −\frac{∂F}{∂x_{j}} (x, y, λ), 1 ≤ j ≤ n
λ̇_{l} = μ_{l}(x, y, λ), 1 ≤ l ≤ k

is locally integrable over the surface \{(x, y, λ); \frac{∂F}{∂λ_{l}} (x, y, λ) = 0, 1 ≤ l ≤ k\}. Q.E.D.

Problem 8.1 Find conditions to be posed on a smooth generating family F : R^{2n}×R^{k} → R, so that the linear equation (10) has a smooth solution on C_{F} .

Let E_{s}(k, k) denote the space of k × k symmetric matrices of real numbers. For each integer r ≥ 0 let Σ_{r} denote the subset of E_{s}(k, k) consisting of all symmetric matrices of rank r. Then Σ_{r} is a submanifold of E_{s}(k, k) of codimension (k − r)(k − r + 1)/2. Now we have a well defined mapping of M_{F} into symmetric matrices E_{s}(k, k). Because the projections π_{T R^{2n}} |M̃_{F}: M̃_{F} → M_{F} and π |M̃_{F}: M̃_{F} → C_{F} are embeddings, then we can uniquely represent this mapping by H : C_{F} → E_{s}(k, k),

H(x, y, λ) |_{\{(x,y,λ)∈C_{F} \}}= ( \begin{matrix}\frac{∂^{2}F}{∂λ_{1}∂λ_{1}} (x, y, λ) & … & \frac{∂^{2}F}{∂λ_{1}∂λ_{k}} (x, y, λ) \\ ⋮ & ⋮ & ⋮ \\ \frac{∂^{2}F}{∂λ_{k}∂λ_{1}} (x, y, λ) & … & \frac{∂^{2}F}{∂λ_{k}∂λ_{k}} (x, y, λ) \end{matrix} )  (13)

Definition 8.2 An implicit Hamiltonian system M_{F} ⊂ T R^{2n} ≡ R^{2n} × R^{2n}, generated by generating family F : R^{2n} × R^{k} → R is called generic if the map H : C_{F} → E_{s}(k, k) is transversal to all Σ_{r}, r = 0, …, k − 1.

Now we can formulate the main result of this section.

Theorem 8.1 The generic implicit Hamiltonian system M ⊂ T R^{2n} is integrable if and only if it satisfies the N-integrability condition.

Proof. Proof of this theorem follows from the J. Mather’s theorem (cf. [12], p. 185) concerning the smooth solvability of linear equations with symmetric matrix. Q.E.D.

Using Section 7 we can formulate the corresponding results concerning integrability of implicit Hamiltonian systems. Now our function-matrix H |_{{(x,y,λ)∈C_{F} }}: C_{F} → E_{s}(k, k) corresponds to matrix A(x) in Proposition 7.4. So on the basis of Propositions 7.1 - 7.4 we get the result corresponding to Theorem 7.1.

Theorem 8.2 Let F : (R^{2n} × R^{k}, 0) → R be a real analytic function-germ. Let M_{F} generated by F fulfills the N-integrability condition. If det(H |_{C_{F}})(x, y, λ) satisfies one of the conditions I or II in Proposition 7.4, then the germ at (0, 0) of M_{F} is integrable.

The class of implicit Hamiltonian systems for which the N-integrability condition is also the sufficient condition is provided by the generating family germs of the form

F (x, y, λ) = f(x, y) + ∑_{i=1}^{k} λ_{i}f_{i}(x, y),

where f̃(x, y) = (f_{1}(x, y), …, f_{k}(x, y)), 1 ≤ k ≤ is a nonsingular smooth map-germ R^{2n}, 0 → R^{k}, 0. Thus M_{F} , called the constrained Hamiltonian system, is a smooth Lagrangian submanifold germ over the submanifold K = f̃^{−1}(0).

Now the N-integrability condition from the Lemma 8.2 becomes the system of equations.

\{\frac{∂F}{∂λ_{i}} , F \} |_{f̃^{−1}(0)}= 0, 1 ≤ i ≤ k  (14)

This system is equivalent to the following one.

{f_{i}(x, y), f_{j}(x, y)} |_{f̃^{−1}(0)}= 0, 1 ≤ i, j ≤ k,
{f(x, y), f_{l}(x, y)} |_{f̃^{−1}(0)}= 0, 1 ≤ l ≤ k,

which in fact defines K to be a coisotropic submanifold of (R^{2n}, ω), i.e. geometrically T_{q}K ⊃ {u ∈ T_{q}R^{2n} | ω(u, v) = 0, ∀_{v∈T_{q}K}}, and f restricts to those who are constant on the leaves of the characteristic filiation of K. Now we have the integrability result proved also in geometric way in ([13], Theorem 8.1).

Corollary 8.2 Let M_{F} be a constrained Hamiltonian system over the submanifold K ⊂ R^{2n} and F fulfills the N-integrability condition. Then K is a coisotropic submanifold of (R^{2n}, ω) and M_{F} is integrable.

9 Integrability on submanifolds

In the preceding sections we studied the sufficient conditions on the existence of smooth solutions of Linear Equations which imply the integrability of submanifolds M of the tangent bundle T R^{n}, where the resultant family of integral curves (γ(t), γ′(t)) were smooth on the whole manifold M. However according to the definition of integrability, the family of integral curves need not to be smooth on the whole manifold M. In this section we observe the cases where there exist no smooth family of integral curves on the whole manifold M but there exists a submanifold N of M on which there exists a smooth family of integral curves. Now we have the basic Lemma.

Lemma 9.1 Let g : R^{m} → R^{n} be a smooth map tangent to a smooth map f : R^{m} → R^{n}. If there exists a submanifold S of R^{m} such that the linear equation

( \begin{matrix}\frac{∂f_{1}}{∂u_{1}} (u) & & & … & & \frac{∂f_{1}}{∂u_{m}} (u) \\ \frac{∂f_{2}}{∂u_{1}} (u) & & & … & & \frac{∂f_{2}}{∂u_{m}} (u) \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ \frac{∂f_{n}}{∂u_{1}} (u) & & & … & & \frac{∂f_{n}}{∂u_{m}} (u) \end{matrix} ) ( \begin{matrix}a_{1}(u) \\ a_{2}(u) \\ … \\ a_{m}(u) \end{matrix} ) = ( \begin{matrix}g_{1}(u) \\ g_{2}(u) \\ … \\ g_{n}(u) \end{matrix} ) , u ∈ S

has a smooth solution tangent to S, i.e

(a_{1}(u), a_{2}(u), …, a_{n}(u)) ∈ T_{u}(S), u ∈ S,

then the submanifold N = (f, g)(S) is an integrable submanifold of T R^{n}. In particular, points of N = (f, g)(S) are integrable points of M = (f, g)(R^{m}).

Proof Since the smooth solution

(a_{1}(u), a_{2}(u), …, a_{n}(u)) ∈ T_{u}(S), u ∈ S,

is a tangent vector field of S, for any point

(x, x′) = (f, g)(u) ∈ N = (f, g)(S),

there exists an integral curve

α : (−ϵ, ϵ) → S such that α′(t) = (a_{1}(α(t)), …, a_{n}(α(t))) and α(0) = u.

Then the curve

γ : (−ϵ, ϵ) → R^{n}, γ(t) = f(α(t))

satisfies

(γ(t), γ′(t)) ∈ N = (f, g)(S), and (γ(0), γ′(0)) = (x, x′) = (f, g)(u).

Thus, the point (x, x′) = (f, g)(u) ∈ N = (f, g)(S) is an integrable point of N = (f, g)(S) and hence N = (f, g)(S) is an integrable submanifold of T R^{n}. Q.E.D.

Corollary 9.1 Let g : R^{m} → R^{n} be a smooth map tangent to a smooth map f : R^{m} → R^{n}. If there exists an embedding ϕ : R^{k} → R^{m} such that the linear equation

J(f ∘ ϕ)(v)b(v) = g(ϕ(v)), v ∈ R^{k}

has a smooth solution b(v), then N = ((f, g) ∘ ϕ)(R^{k}) is an integrable submanifold of T R^{n}. In particular, points of N = ((f, g)∘ϕ)(R^{k}) are integrable points of M = (f, g)(R^{m}).

Proof Set

S = ϕ(R^{k}) and a(ϕ(v)) = Jϕ(b(v))

and apply Fundamental Lemma 9.1 to S and a(u = ϕ(v)). Q.E.D.

Now we describe a type of examples which are the generalization of Example 6.2. Let M be the image of an embedding of R^{m}:

M = (f, g)(R^{m}), (f, g) : R^{m} → T R^{n} = R^{n} × R^{n} an embedding.

Suppose the Jacobian matrix Jf(0) at 0 ∈ R^{m} of f has rank r:

rankJf(0) = r.

Then there exists a local coordinate system in a neighborhood of the origin of R^{m}

(v, w) = (v_{1}, …, v_{r}, w_{r+1}, …, w_{m})

such that

f(v, w) = (v_{1}, …, v_{r}, f_{r+1}(v, w), …, f_{n}(v, w)).

Let us denote

E_{v,w} = the ring of C^{∞} function germs at (u, v) = (0, 0)
of the variables v_{1}, …, v_{r}, w_{r+1}, …, w_{m},
E_{w} = the ring of C^{∞} function germs at w = 0
of the variables w_{r+1}, …, w_{m},
M_{w} = the maximal ideal of E_{w}

Theorem 9.1 Let M = (f, g)(R^{m}) be the image of the germ at 0 ∈ R^{m} of an embedding (f, g) : R^{m} → T R^{n} satisfying N-integrability condition. Let

rankJf(0) = r

and let (v, w) = (v_{1}, …, v_{r}, w_{r+1}, …, w_{m}) be a local coordinate system in a neighborhood of the origin of R^{m} such that

f(v, w) = (v_{1}, …, v_{r}, f_{r+1}(v, w), …, f_{n}(v, w)).

Let

S = {(v, w) ∈ R^{m} | w_{r+1} = w_{r+2} = ⋯ = w_{m} = 0} = R̄^{r} × {0}.

If

f_{r+1}(v, w), …, f_{n}(v, w) ∈ M_{w}^{2}E_{v,w},

then N = (f, g)(S) is an integrable submanifold of T R^{n}.

Proof Since

f_{r+1}(v, w), …, f_{n}(v, w) ∈ M_{w}^{2}E_{v,w},

we see that the Jacobian matrix Jf(v, 0) has the form

Jf(v, 0) = ( \begin{matrix}I_{r} & O \\ O & O \end{matrix} ) ,

where I_{r} is the identity matrix of size r, and the linear equation has the form

g(v, 0) = ( g_{1}(v, 0) ; ⋮ ; g_{n}(v, 0) ) = ( \begin{matrix}I_{r} & O \\ O & O \end{matrix} ) ( a_{1}(v, 0) ; ⋮ ; a_{m}(v, 0) ) .

Therefore we have

g(v, 0) = ( g_{1}(v, 0) ; ⋮ ; g_{r}(v, 0) ; 0 ; ⋮ ; 0 ) = ( \begin{matrix}I_{r} & O \\ O & O \end{matrix} ) ( g_{1}(v, 0) ; ⋮ ; g_{r}(v, 0) ; 0 ; ⋮ ; 0 ) = Jf(v, 0) ( g_{1}(v, 0) ; ⋮ ; g_{r}(v, 0) ; 0 ; ⋮ ; 0 ) .

Thus the linear equation

g(u) = Jf(u)a(u), u ∈ S = {u = (v, 0)}

has a smooth solution

(a_{1}(v, 0), …, a_{n}(u, 0)) = (g_{1}(v, 0), …, g_{r}(v, 0), 0, …, 0)

tangent to S. Therefore, from Fundamental Lemma 9.1, N = (f, g)(S) is an integrable submanifold of T R^{n}. Q.E.D.

Now we consider the relation between integral curves at regular points of π_{1} | M → R^{n} and the integral curves on the submanifold S given by Theorem 9.1 .

Remark 9.1 Let us define

Σ = {(v, w) ∈ R^{m} | rankJf(v, w) < min{m, n} }.

Then we have

S ⊂ Σ.

Since all the points

(x, x′) ∈ M − (f, g)(Σ)

are regular points of the projection

π_{1} : T R^{n} = R^{n} × R^{n} → R^{n}, π_{1}(x, x′) = x,

from Lemma 2.2, M − (f, g)(Σ) is an integrable submanifold of T R^{n}. Concerning the integrability of points of the set (f, g)(Σ) − N , we need more precise observation, which will be done in the forthcoming paper on Stratification of singular point sets and N-Integrability conditions.

Remark 9.2 Let M be the image of an embedding (f, g) : R^{m} → T R^{n} = R^{n} × R^{n}. Moreover suppose that f : R^{m} → R^{n} be of the form

f(v, w) = (v_{1}, …, v_{r}, f_{r+1}(v, w), …, f_{n}(v, w)).

In this situation, if

g_{r+1}(v, w), …, g_{n}(v, w) ∈ M_{w}E_{v,w},

then M = (f, g)(R^{m}) satisfies N-integrability condition. However if

f_{r+1}(v, w), …, f_{n}(v, w) ∈ M_{w}^{3}E_{v,w}

and if

g_{r+1}(v, w), …, g_{n}(v, w) ∈ M_{w}E_{v,w} − M_{w}^{2}E_{v,w},

then the linear equation

g(v, w) = Jf(v, w)a(v, w)

has no smooth solution on the whole R^{m}, while it has a smooth solution b(v, 0) defined on S = {(v, w) ∈ R^{m} | w = 0};

g(v, 0) = Jf(v, 0)b(v, 0), (v, 0) ∈ S,

and also a smooth solution c(v, w) on R^{m} − Σ;

g(v, w) = Jf(v, w)c(v, w), (v, w) ∉ Σ,
where Σ = {(v, w) ∈ R^{m} | rankJf(v, w) < min{m, n} }.

Let us now observe the solution c(v, w) at points (v, w), which are very close to S. The Jacobian matrix Jf(v, w) has the form

Jf(v, w) = ( I_{r}  O_{\frac{∂f_{r+i}}{∂v_{j}}  \frac{∂f_{r+i}}{∂w_{r+k}}} )_{1≤i≤n−r, 1≤j≤r, 1≤k≤m−r} ,

Therefore the first r solutions c_{1}(v, w), …, c_{r}(v, w) has the form

c_{1}(v, w) = g_{1}(v, w), …, c_{r}(v, w) = g_{r}(v, w).

The rest solutions c_{r+1}(v, w), …, c_{m}(v, w) must satisfy the equality

( g_{r+1}(v, w) ; ⋮ ; g_{n}(v, w) ) = ( \begin{matrix}\frac{∂f_{r+1}}{∂v_{1}} & & ⋯ & & \frac{∂f_{r+1}}{∂v_{r}} & & & \frac{∂f_{r+1}}{∂w_{r+1}} & & ⋯ & & \frac{∂f_{r+1}}{∂w_{m}} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ \frac{∂f_{n}}{∂v_{1}} & & ⋯ & & \frac{∂f_{n}}{∂v_{r}} & & & \frac{∂f_{n}}{∂w_{r+1}} & & ⋯ & & \frac{∂f_{n}}{∂w_{m}} \end{matrix} ) ( g_{1}(v, w) ; ⋮ ; g_{r}(v, w) ; c_{r+1}(v, w) ; ⋮ ; c_{m}(v, w) ) .

Therefore they must satisfy the equality

( g_{r+1}(v, w) ; ⋮ ; g_{n}(v, w) ) − ( \begin{matrix}\frac{∂f_{r+1}}{∂v_{1}} & & & ⋯ & & & \frac{∂f_{r+1}}{∂v_{r}} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ \frac{∂f_{n}}{∂v_{1}} & & & ⋯ & & & \frac{∂f_{n}}{∂v_{r}} \end{matrix} ) ( g_{1}(v, w) ; ⋮ ; g_{r}(v, w) )

= ( \begin{matrix}\frac{∂f_{r+1}}{∂w_{r+1}} & & & ⋯ & & \frac{∂f_{r+1}}{∂w_{m}} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ \frac{∂f_{n}}{∂w_{r+1}} & & & ⋯ & & \frac{∂f_{n}}{∂w_{m}} \end{matrix} ) ( c_{r+1}(v, w) ; ⋮ ; c_{m}(v, w) ) .

Since

f_{r+1}(v, w), …, f_{n}(v, w) ∈ M_{w}^{3}E_{v,w},

and

g_{r+1}(v, w), …, g_{n}(v, w) ∈ M_{w}E_{v,w} − M_{w}^{2}E_{v,w},

comparing the orders with respect to w_{r+1}, …, w_{m} of the both hand sides of the equation, we see

lim_{∥w∥→0} c_{r+i}(v, w) = ∞, i = 1, …, m − r.

On the other hand the solution b(v, 0) has the form

(b_{1}(v, 0), …, b_{n}(v, 0)) = (g_{1}(v, 0), …, g_{r}(v, 0), 0, …, 0).

Thus

lim_{∥w∥→0}^{}(c(v, w), b(v, 0)) = \frac{π}{2} ,

where ^{}(α, β) denotes the angle of the vectors α and β.

Thus we can never have a smooth solution of the linear equation

g(v, w) = Jf(v, w)a(v, w)

by glueing c(v, w) with b(v, 0).

Now we investigate the codimension 1 singularities. For simplicity, we consider the case m = n. Let M be the image of an embedding of R^{m}:

M = (f, g)(R^{m}), (f, g) : R^{m} → T R^{m} = R^{m} × R^{m} an embedding.

Suppose the Jacobian matrix Jf(0) at 0 ∈ R^{m} of f has rank m − 1:

rankJf(0) = m − 1.

Then there exists a local coordinate system in a neighborhood of the origin of R^{m}

(v, w) = (v_{1}, …, v_{m−1}, w_{m})

such that

f(v, w) = (v_{1}, …, v_{m−1}, f_{m}(v, w)).

Theorem 9.2 Let M = (f, g)(R^{m}) be the image of the germ at 0 ∈ R^{m} of an embedding (f, g) : R^{m} → T R^{m} satisfying N-integrability condition. Let

rankJf(0) = m − 1

and let (v, w) = (v_{1}, …, v_{m−1}, w_{m}) be a local coordinate system in a neighborhood of the origin of R^{m} such that

f(v, w) = (v_{1}, …, v_{m−1}, f_{m}(v, w)).

Suppose that

detJf(v, w) = \frac{∂f_{m}}{∂w_{m}} (v, w)

has the form

detJf(v, w) = (h_{1}(v, w))^{k}h_{2}(v, w) with \frac{∂h_{1}}{∂w_{m}} (0, 0) ≠ 0.

Set

S = {(v, w) ∈ R^{m} | h_{1}(v, w) = 0}, N = (f, g)(S).

Then N is an integrable submanifold of T R^{m}.

Proof Consider the linear equation

g(v, w) = Jf(v, w)a(v, w), (v, w) ∈ S.

We are going to construct a smooth solution a(v, w), (v, w) ∈ S of the equation tangent to S. Then from Fundamental Lemma 9.1, we see that S is an integrable submanifold of T R^{m}.

Since Jf(v, w) has the form

Jf(v, w) = ( \begin{matrix}I_{m−1} & O \\ \frac{∂f_{m}}{∂v_{j}} (v, w) & (h_{1}(v, w))^{k}h_{2}(v, w) \end{matrix} ) ,

we see that

Jf(v, w) = ( \begin{matrix}I_{m−1} & O \\ \frac{∂f_{m}}{∂v_{j}} (v, w) & 0 \end{matrix} ) , ∀(v, w) ∈ S.

Therefore first of all, g_{m}(v, w) must satisfy the equality

g_{m}(v, w) = ∑_{j=1}^{m−1} \frac{∂f_{m}}{∂v_{j}} (v, w)g_{j}(v, w), ∀(v, w) ∈ S.

And if we choose a_{1}(v, w), …, a_{m−1}(v, w) as

a_{1}(v, w) = g_{1}(v, w), …, a_{m−1}(v, w) = g_{m−1}(v, w),

then for any function a_{m}(v, w),

a(v, w) = (a_{1}(v, w), …, a_{m−1}(v, w), a_{m}(v, w)) = (g_{1}(v, w), …, g_{m−1}(v, w), a_{m}(v, w))

is a solution of the equation

g(v, w) = Jf(v, w)a(v, w), (v, w) ∈ S.

Now we choose a_{m}(v, w) so that the vector field a(v, w) is tangent to S. Consider the gradient vector field of h_{1}(v, w):

gradh_{1}(v, w) = (\frac{∂h_{1}}{∂v_{1}} (v, w), …, \frac{∂h_{1}}{∂v_{m−1}} (v, w), \frac{∂h_{1}}{∂w_{m}} (v, w)).

Choose a function a_{m}(v, w), (u, v) ∈ S, so that the vector field

(g_{1}(v, w), …, g_{m−1}(v, w), a_{m}(v, w))

is orthogonal to gradh_{1}(v, w):

g_{1}(v, w)\frac{∂h_{1}}{∂v_{1}} (v, w) + ⋯ + g_{m−1}(v, w)\frac{∂h_{1}}{∂v_{m−1}} (v, w) + \frac{∂h_{1}}{∂w_{m}} (v, w)a_{m}(v, w) = 0.

From the assumption that \frac{∂h_{1}}{∂w_{m}} (0, 0) ≠ 0, we have such a_{m}(v, w), namely

a_{m}(v, w) = \frac{−1}{\frac{∂h_{1}}{∂w_{m}} (v, w)} ∑_{j=1}^{m−1} g_{j}(v, w)\frac{∂h_{1}}{∂v_{j}} (v, w).

Then, with the coefficient function a_{m}(v, w), the vector field

(a_{1}(v, w), …, a_{m}(v, w)) = (g_{1}(v, w), …, g_{m−1}(v, w), a_{m}(v, w)) (v, w) ∈ S

is tangent to S. In fact (a_{1}(v, w), …, a_{m}(v, w)) is orthogonal to the gradient vector field gradh_{1}(v, w) and h_{1}(v, w) is the defining function of S. Thus a(v, w), (v, w) ∈ S is a smooth solution of the linear equation

g(v, w) = Jf(v, w)a(v, w), (v, w) ∈ S.

tangent to S. Now, from Fundamental Lemma 9.1, we see that S is an integrable submanifold of T R^{m}. Q.E.D.

In the same way as Theorem 9.2, we can prove the following Corollary 9.2 Let M = (f, g)(R^{m}) be the image of the germ at 0 ∈ R^{m} of an embedding (f, g) : R^{m} → T R^{m} satisfying N-integrability condition. Let

rankJf(0) = m − 1

and let (v, w) = (v_{1}, …, v_{m−1}, w_{m}) be a local coordinate system in a neighborhood of the origin of R^{m} such that

f(v, w) = (v_{1}, …, v_{m−1}, f_{m}(v, w)).

Suppose that

detJf(v, w) = \frac{∂f_{m}}{∂w_{m}} (v, w)

has the form

detJf(v, w) = h_{1}(v, w)^{k_{1}}h_{2}(v, w)^{k_{2}} ⋯ h_{s}(v, w)^{k_{s}}h(v, w)

with \frac{∂h_{i}}{∂w_{m}} (0, 0) ≠ 0, i = 1, …, s.

Set

S_{i} = {(v, w) ∈ R^{m} | h_{i}(v, w) = 0}, and N_{i} = (f, g)(S_{i}).

Then N_{i}’s are integrable submanifolds of T R^{m}.

We can prove similar results for the cases m ≠ n. However we leave it to the forthcoming paper.

10 Non-integrable points are generically isolated

In this section we will investigate the non integrable points. We prove that non-transversal points, i.e. the points where Jf : R^{m} → E(n, m) is not transversal to S_{r}, are generically isolated, which implies that non-integrable points are generically isolated.

Here, in this section, the genericity is in the strongest sense of Tougeron [20]:

Definition 10.1 We say that the non-transversal points of M = (f, g)(R^{m}) are generically isolated if there exists an ∞ codimensional subset Σ_{∞}(R^{m}, R^{n}) of the space C^{∞}(R^{m}, R^{n}) of C^{∞} mappings of R^{m} into R^{n} such that for any f ∈ C^{∞}(R^{m}, R^{n}) − Σ_{∞}(R^{m}, R^{n}) nontransversal points are isolated.

Let us fix the notation now. Set

C^{∞}(R^{m}, T R^{n}) = {(f, g) : R^{m} → R^{n} × R^{n} = T R^{n} | C^{∞} mapping}
= C^{∞}(R^{m}, R^{n}) × C^{∞}(R^{m}, R^{n})
Emb(R^{m}, T R^{n}) = {(f, g) : R^{m} → R^{n} × R^{n} = T R^{n} | embedding},
N (R^{m}, T R^{n}) = {(f, g) ∈ C^{∞}(R^{m}, T R^{n}) | (f, g) satisfies N-integrability
condition g(u) ∈ Jf(u)(T_{u}(R^{n})) ∀u ∈ R^{m}}
N Int(R^{m}, T R^{n}) = Emb(R^{m}, T R^{n}) ∩ N (R^{m}, T R^{n}),

where all these spaces are endowed with Whitney C^{∞} topology, and Jf(u) is the Jacobian matrix of f at u ∈ R^{m}.

Definition 10.2 Let (f, g) ∈ N Int(R^{m}, T R^{n}). A point (x, ẋ) = (f(u), g(u)) ∈ M = (f, g)(R^{m}) is a transversal point of M if Jf : R^{m} → E(n, m) is transversal to S_{r} at u ∈ R^{m} for r = rankJf(u), equivalently j^{1}(π_{1} |_{M}) : M → J^{1}(M, R^{n}) is transversal to

S_{r}(M, R^{n}) = {j^{1}h(v) ∈ J^{1}(M, R^{n}) | rankJh(v) = r},

where π_{1} : T R^{n} = R^{n} × R^{n} → R^{n} is the projection defined by π_{1}(x, ẋ) = x.

We are interested in the structure of the non-generic (in the sense of the previous sections) points and to investigate how they are located in M. We prove the following main theorems.

Theorem 10.1 There exists an ∞ codimensional subset Σ_{∞}N Int(R^{m}, T R^{n}) of N Int(R^{m}, T R^{n}) such that for any embedding

(f, g) ∈ N Int(R^{m}, T R^{n}) − Σ_{∞}N Int(R^{m}, T R^{n})

non-transversal points of M = (f, g)(R^{m}) are isolated.

Combining this theorem with J. Mather’s theorem, we obtain

Theorem 10.2 Let Σ_{∞}N Int(R^{m}, T R^{n}) be the ∞ codimensional subset of N Int(R^{m}, T R^{n}) given in Theorem 10.1. Then for any

(f, g) ∈ N Int(R^{m}, T R^{n}) − Σ_{∞}N Int(R^{m}, T R^{n})

non-integrable points of M = (f, g)(R^{m}) are isolated.

Remark 10.1 Remark Let

Σ_{tr}N Int(R^{m}, T R^{n}) = {(f, g) ∈ N Int(R^{m}, T R^{n}) | Jf : U → E(n, m)

is not transversal to S_{r} for some r}.

Then we see

Σ_{∞}N Int(R^{m}, T R^{n}) ⊂ Σ_{tr}N Int(R^{m}, T R^{n}),
codimΣ_{tr}N Int(R^{m}, T R^{n}) in N Int(R^{m}, T R^{n}) < ∞,
codimΣ_{∞}N Int(R^{m}, T R^{n}) in N Int(R^{m}, T R^{n}) = ∞.

This means that even if M = (f, g)(R^{m}) is not a generic submanifold in the sense of section 3, non-transversal points and non-integrable points of M = (f, g)(R^{m}) are isolated points for a very generic (f, g), i.e. for any (f, g) ∈ Σ_{tr}N Int(R^{m}, T R^{n}) − Σ_{∞}N Int(R^{m}, T R^{n}).

For the strong genericity result we need an appropriate transversality theorem. Let us denote

J^{r}(m, n) = {j^{r}f(0) | f ∈ C^{∞}(R^{m}, R^{n}) and f(0) = 0}
J^{r}(R^{m}, R^{n}) = {j^{r}f(u) | f ∈ C^{∞}(R^{m}, R^{n})), u ∈ R^{m}}.

For positive integers s > r > 0, we define the canonical projections

π_{r}^{s} : J^{s}(m, n) → J^{r}(m, n), π_{r}^{s}(j^{s}f(0)) = j^{r}f(0).

Theorem 10.3 (Transversality) Let W be a semi-algebraic subset of J^{r}(m, n) and let X_{1}, …, X_{ℓ} be a finite number of semi-algebraic submanifolds of J^{k}(R^{m}, R^{n}). Then there exists a closed semi-algebraic subset

Σ_{W} of (π_{r}^{r+k+1})^{−1}(W) having codimension ≥ 1

such that for any C^{∞} mapping f : R^{m} → R^{n} with

j^{r+k+1}(0) ∈ (π_{r}^{r+k+1})^{−1}(W) − Σ_{W} ,

there exists a neighborhood U of the origin 0 of R^{m} such that j^{r}f : U −{0} → J^{k}(R^{m}, R^{n}) is transversal to X_{1}, …, X_{ℓ} at every point u ∈ U except at the origin 0 ∈ R^{m}

For the proof we refer to the more general version of Theorem 10.3 which is proved in [7]. (Theorem 1 in [7], p. 229)

10.1 Proof of Theorem 10.1

Let us identify J^{1}(m, n) with E(n., m) and set

S_{r}(R^{m}, R^{n}) = {j^{1}f(u) ∈ J^{1}(R^{m}, R^{n}) | rankJf(u) = r}
= R^{m} × R^{n} × S_{r} ⊂ R^{m} × R^{n} × J^{1}(m, n) = J^{1}(R^{m}, R^{n}).

For (u_{0}, x_{0}) ∈ R^{m} × R^{n}, set

C^{∞}(R^{m}, R^{n}; u_{0}, x_{0}) = {f ∈ C^{∞}(R^{m}, R^{n})) | f(u_{0}) = x_{0}}.

We prove Main Theorem 10.1 using the following result

Theorem 10.4 There exists an infinite codimensional subset Σ_{∞}(R^{m}, R^{n}; u_{0}, x_{0}) of C^{∞}(R^{m}, R^{n}; u_{0}, x_{0}) such that for any C^{∞} mapping f : R^{m} → R^{n} with

f ∈ C^{∞}(R^{m}, R^{n}; u_{0}, x_{0}) − Σ_{∞}(R^{m}, R^{n}; u_{0}, x_{0})

there exists a neighborhood V (u_{0}) of u_{0} in R^{m} such that j^{1}f : V (u_{0})−{u_{0}} → J^{1}(R^{m}, R^{n}) is transversal to S_{r}(R^{m}, R^{n}) for ∀r = 0, …, min(m, n) at every point u ∈ V (u_{0}) − {u_{0}} except at u_{0}.

Proof of Theorem 10.1 Set

Σ_{∞}(R^{m}, R^{n}) = ∪_{(u,x)∈R^{m}×R^{n}}Σ_{∞}(R^{m}, R^{n}; u, x) ⊂ C^{∞}(R^{m}, R^{n})
Σ_{∞}(R^{m}, T R^{n}) = {(f, g) ∈ C^{∞}(R^{m}, T R^{n}) | f ∈ Σ_{∞}(R^{m}, R^{n})}
= Σ_{∞}(R^{m}, R^{n}) × C^{∞}(R^{m}, R^{n})
Σ_{∞}N Int(R^{m}, T R^{n}) = Σ_{∞}(R^{m}, T R^{n}) ∩ N Int(R^{m}, T R^{n}),

where we note that T R^{n} = R^{n} × R^{n} and hence that

C^{∞}(R^{m}, T R^{n}) = C^{∞}(R^{m}, R^{n}) × C^{∞}(R^{m}, R^{n}).

We claim that Σ_{∞}N Int(R^{m}, T R^{n}) is the required one.

It is easy to see that it satisfies the condition: if (f, g) ∈ N Int(R^{m}, T R^{n}) − Σ_{∞}N Int(R^{m}, T R^{n}), then non-transversal points of M = (f, g)(R^{m}) are isolated.

Now we investigate the codimension of this set. Since Σ_{∞}(R^{m}, R^{n}; u_{0}, x_{0}) is ∞ codimensional in C^{∞}(R^{m}, R^{n}; u_{0}, x_{0}), so is Σ_{∞}(R^{m}, R^{n}) in C^{∞}(R^{m}, R^{n}). Therefore

codimΣ_{∞}(R^{m}, R^{n}) × C^{∞}(R^{m}, R^{n}) in C^{∞}(R^{m}, T R^{n}) is equal to ∞.

Recall that we defined

N (R^{m}, T R^{n}) = {(f, g) ∈ C^{∞}(R^{m}, T R^{n}) | g(u) ∈ Jf(u)(R^{m}), ∀u ∈ R^{m}}.

and

N Int(R^{m}, T R^{n}) = Emb(R^{m}, T R^{n}) ∩ N (R^{m}, T R^{n}).

Since

Σ_{∞}(R^{m}, T R^{n}) = Σ_{∞}(R^{m}, R^{n}) × C^{∞}(R^{m}, R^{n})

and

codimΣ_{∞}(R^{m}, R^{n}) × C^{∞}(R^{m}, R^{n}) in C^{∞}(R^{m}, T R^{n}) is equal to ∞,

we have

codimΣ_{∞}(R^{m}, T R^{n}) ∩ N (R^{m}, T R^{n}) in N (R^{m}, T R^{n}) is equal to ∞.

Since

Emb(R^{m}, T R^{n}) is open in C^{∞}(R^{m}, T R^{n}),

we see that

codimΣ_{∞}(R^{m}, T R^{n}) ∩ N (R^{m}, T R^{n}) ∩ Emb(R^{m}, T R^{n})
in N (R^{m}, T R^{n}) ∩ Emb(R^{m}, T R^{n}) is equal to ∞.

Since

Σ_{∞}N Int(R^{m}, T R^{n}) = Σ_{∞}(R^{m}, T R^{n}) ∩ N Int(R^{m}, T R^{n})
= Σ_{∞}(R^{m}, T R^{n}) ∩ N (R^{m}, T R^{n}) ∩ Emb(R^{m}, T R^{n}),
N Int(R^{m}, T R^{n}) = N (R^{m}, T R^{n}) ∩ Emb(R^{m}, T R^{n})

we see that

codimΣ_{∞}N Int(R^{m}, T R^{n}) in N Int(R^{m}, T R^{n}) is equal to ∞.

This completes the proof of Theorem 10.1.

Proof of Theorem 10.4

It suffices to prove Theorem 10.4 for (u_{0}, x_{0}) = (0, 0), so we reformulate it in an accessible way.

Theorem 10.4 There exists an infinite codimensional subset Σ_{∞}(R^{m}, R^{n}; 0, 0) of C^{∞}(R^{m}, R^{n}; 0, 0) such that for any C^{∞} mapping f : R^{m} → R^{n} with

f ∈ C^{∞}(R^{m}, R^{n}; 0, 0) − Σ_{∞}(R^{m}, R^{n}; 0, 0)

there exists a neighborhood V (0) of 0 in R^{m} such that j^{1}f : V (0) − {0} → J^{1}(R^{m}, R^{n}) is transversal to S_{r}(R^{m}, R^{n}) for ∀r = 0, …, min(m, n) at every point u ∈ V (0)−{0} except at the origin 0.

Proof. First we apply Theorem 10.3 to

W = W_{1} = J^{1}(m, n) and
X_{r} = S_{r}(R^{m}, R^{n}) ⊂ J^{1}(R^{m}, R^{n}), r = 0, …, min(m, n).

Then there exists a closed semi-algebraic subset

Σ_{1} of (π_{1}^{1+1+1})^{−1}(W) = J^{3}(m, n) having codimension ≥ 1

such that for any C^{∞} mapping f : R^{m} → R^{n} with

j^{3}(0) ∈ J^{3}(m, n) − Σ_{1},

there exists a neighborhood U of the origin 0 of R^{m} such that

j^{1}f : U − {0} → J^{1}(R^{m}, R^{n}) is transversal to S_{r}(R^{m}, R^{n}), ∀r.

.

Next we apply Theorem 10.3 to

W = Σ_{1} ⊂ J^{3}(m, n) and
X_{r} = S_{r}(R^{m}, R^{n}) ⊂ J^{1}(R^{m}, R^{n}), r = 0, …, min(m, n).

Then, again there exists a closed semi-algebraic subset

Σ_{2} of (π_{3}^{3+1+1})^{−1}(Σ_{1}) ⊂ J^{5}(m, n)

such that

codimension of Σ_{2} in (π_{3}^{5})^{−1}(Σ_{1}) ≥ 1
codimension of Σ_{2} in J^{5}(m, n) ≥ 2

and such that for any C^{∞} mapping f : R^{m} → R^{n} with

j^{5}(0) ∈ J^{5}(m, n) − Σ_{2},

there exists a neighborhood U of the origin 0 of R^{m} such that

j^{1}f : U − {0} → J^{1}(R^{m}, R^{n}) is transversal to S_{r}(R^{m}, R^{n}), ∀r = 0.

In this way, we can prove inductively that there exists a closed semi-algebraic subset

Σ_{k} of J^{2k+1}(m, n) with codimension ≥ k

such that for any C^{∞} mapping f : R^{m} → R^{n} with

j^{2k+1}(0) ∈ J^{2k+1}(m, n) − Σ_{k},

there exists a neighborhood U of the origin 0 of R^{m} such that

j^{1}f : U − {0} → J^{1}(R^{m}, R^{n}) is transversal to S_{r}(R^{m}, R^{n}), ∀r.

Set

Σ_{k}(R^{m}, R^{n}; 0, 0) = {f ∈ C^{∞}(R^{m}, R^{n}; 0, 0) | j^{2k+1}(0) ∈ Σ_{k}},
Σ_{∞}(R^{m}, R^{n}; 0, 0) = ∩_{k=1}^{∞}Σ_{k}(R^{m}, R^{n}; 0, 0).

Then Σ_{∞}(R^{m}, R^{n}; 0, 0) satisfies the condition required in Theorem 4. This completes the proof of Theorem 10.4 and hence also the proof of Main Theorem 10.1. Q.E.D.

References

[1] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of Differentiable Maps, Vol. 1, Birhauser, Boston, 1985.

[2] J. Basto-Goncalves, Implicit Hamiltonian equations, Mat. Contemp., 12, (1997), 123-137

[3] J.W. Bruce, A note on first order differential equations of degree greater than one and wavefront evolution, Bull. London Math. Soc. 16 (1984), 139-144.

[4] A. Davydov, Normal form for a differential equation, not solvable for the derivative, in a neighbourhood of a singular point, Funct. Anal. Appl., 19, (1985), 81-89.

[5] P.A.M. Dirac, Generalized Hamiltonian Dynamics , Canadian J. Math. 2, (1950), 129-148.

[6] I. Ekeland, Discontinuités de champs hamiltoniens et existence de solutions optimales en calcul des variations, Publ. Math. I.H.E.S. 47 (1977), 5-32.

[7] T. Fukuda, Local topological properties of differentiable mappings I, Invent. Math. 65, (1981), 227-250.

[8] C.G. Gibson, K. Wirthmüller, A.A. du Plessis, E.J.N. Loijenga, Topological stability of smooth mappings, Springer Lecture Notes in Mathematics, LNM 552, Springer 1976 [9] S. Janeczko, On implicit Lagrangian differential systems, Annales Polonici Mathe-

matici, LXXIV, (2000),133-141

[10] W. Liu, H.J. Sussmann, Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer. math. Soc. 118 (1995), no.564.

[11] J. Martinet, Singularities of Smooth Functions and Maps, Cambridge Univ. Press, Cambridge, 1982.

[12] J.N. Mather, Solutions of generic linear equations, Dynamical Systems, (1972),185-193.

[13] M.R. Menzio, W.M. Tulczyjew, Infinitesimal symplectic relations and generalized Hamiltonian dynamics, Ann. Inst. H. Poincaré, XXVIII, No. 4, (1978), 349-367.

[14] B. Morin, Formes canoniques des singularités d’une application différentiable, C.R.Acad. Sci. Paris, t. 260 (1965), 5662-5665.

[15] R. Narasimhan, Introduction to the Theory of Analytic space, Lecture Notes in Math. 25, Springer Verlag, 1966.

[16] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mischenko, The Mathematical Theory of Optimal Processes, Pergamon Press, New York-Paris 1964.

[17] J. Sotomayor, M. Zhitomirskii, Impasse singularities of differential systems of the form A(x)x′ = F (x), Journal of Diff. Equations 169, (2001), 567-587.

[18] F. Takens, Implicit differential equations: some open problems, Springer Lecture Notes in Mathematics, Vol. 535, (1976), 237-253.

[19] R. Thom, Sur les équations différentielles multiforms et leurs intégrales singuliérs, Colloque E. Cartan, Paris, 1971.

[20] J.C. Tougeron, Idéaux de fonctions différentiables I, Ann. Inst. Fourier 18-1, (1986), 177-240.

[21] A. Weinstein, Lectures on Symplectic Manifolds, CBMS Regional Conf. Ser. in Math., 29, AMS Providence, R.I. 1977.

Takuo Fukuda Department of Mathematics College of Humanities and Sciences Sakurajousui 3-25-40, Setagaya-ku, Tokyo, Japan

Stanislaw Janeczko Warsaw University of Technology, Pl. Politechniki 1, 00-663 Warsaw, Poland and Mathematical Institute Polish Academy of Sciences, ul. Sniadeckich 8, 00-950 Warsaw, Poland