Injective mappings and solvable vector fields

MATHEMATICAL SCIENCES

Injective mappings and solvable vector fields

José R. dos Santos Filho^I; Joaquim Tavares^II

^IDepartamento de Matemática, Universidade Federal de São Carlos Via Washington Luis, km 235, 13565-905 São Carlos, SP, Brasil

^IIDepartamento de Matemática, Universidade Federal de Pernambuco Av. Prof. Luiz Freire s/n, 50740-540 Recife, PE, Brasil

^{Correspondence to}

ABSTRACT

We establish a sufficient condition for injectivity in a class of mappings defined on open connected subsets of Rⁿ , for arbitrary n. The result relates solvability of the appropriate vector fields with injectivity of the mapping and extends a result proved by the first author for n < 3 . Furthermore, we extend the result to connected paracompact smooth oriented manifolds and show that the convexity condition imposes strong topological restrictions on the manifold.

Key words: fields, injectivity, mappings, solvability, vectors.

RESUMO

Nós estabelecemos uma condição suficiente para injetividade numa classe de aplicações definidas em subconjuntos abertos conexos de Rⁿ , para n arbitrário. O resultado relaciona resolubilidade de campos de vetores apropriados com injetividade da aplicação e estende o resultado demonstrado pelo primeiro autor quando n < 3 . Além disso, nós estendemos o resultado para variedades suaves orientadas e para-compactas e mostramos que a condição de convexidade impõe fortes restrições topológicas na variedade.

Palavras-chave: campos, injetividade, aplicações, resolubilidade, vetores.

INTRODUCTION

Let be an open connected subset of , and consider . We denote an element of by . A very basic problem in Mathematical Analysis is to determine additional conditions under which is injective. The most famous related problem is the so called Jacobian Conjecture, which states that for a polynomial mapping on no additional condition is necessary, see (Bass et al. 1982). For applications and related bibliography see (Santos Filho 2004).As in Santos Filho, for each , we consider the real vector field defined by

Here the mapping is given by: Its j-component is equal to if and its i-component is equal to . It follows that the non-empty connected components of , where , defines a smooth one-dimensional foliation of . This foliation isprecisely that one of the characteristic curves of , because each is a first integral of , if with . In order to characterize solvability of linear partial differential operators on a manifold, in (Malgrange 1956), B. Malgrange introduced a notion of convexity, namely the condition (a)(2) of Theorem 2.1 below. As in (Santos Filho 2004), a similar notion is in order.

DEFINITION . Let . We say that is -convex if there are an open set of of , with , and where, for , each is a diffeomorphism over its image such that there are different indices where is -convex for and .

For it was proved in Theorems 0.1 and Theorem 0.2 of (Santos Filho 2004), that: If is -convex then is injective. Also is simply connected when .

Here we give a full generalization of these results. We mean not only for arbitrary dimensions, but also for the smooth category of oriented manifolds too. Our main goal is to generalize, for higher dimensions and arbitrary manifolds, those results. First we address the euclidean case as a introduction inTheorem 1.1 and then we furnish the needed tools for the general case in Theorem 2.2 below.

THEOREM . Let be an open connected subset of and . If is -convex then is injective.

EXAMPLE. Consider the smooth mapping of the plane given by for and if . Then , so is -convex. Moreover if , so is positive except at a closed set of null Lebesgue measure, in the other hand is not injective. So the condition of in Theorem 1.1 can not be in this fashion.

This paper is organized in the following way: First we prove Theorem 1.1, then extend it for connected paracompact smooth oriented n-dimensional manifold and finally show that must be contractible if is -convex for some . We conclude by making some remarks regarding the results.

PROOFS OF THE RESULTS AND REMARKS

Before we prove our theorems we recall part of Theorem 6.4.2 of Duistermaat and Hörmander, in (Duistermaat and Hörmander 1972), this result characterizes global solvability of vector fields considered as partial differential operators:

THEOREM . Let be a smooth real vector field of a manifold , the following conditions (a) and (b) below are equivalent:

(a) (1) No complete integral curve of is contained in compact subset of .

(2) For every compact subset of there exists a compact subset of such that every compact interval of an integral curve with end points in is contained in .

(b) There exists a manifold and an open neighborhood of in which is convex in the direction, and a diffeomorphism from which carries into the operator if points of are denoted by .

In the first subsection we prove Theorem 1.1, in the second subsection we state and prove an extension of it. Then a topological consequence, as corollary, is deduced and remarks are made.

PROOF OF THEOREM 1.1

Without loss of generality we can assume that

and for . Our prooffollows by a finite induction argument once we prove injectivity for appropriate restrictions of . Let an integer, , and , consider be a connected component of . From the hypotheses, it follows that for each is a submanifold of dimension of , which is invariant by the flow of of , for .

Let , since where is the Kronecker symbol, we have that both -limit and -limit of characteristic curves of are empty. Hence condition (a)(1) of Theorem 2.1 is true for , if . Therefore, in our case, condition (a) of Theorem 2.1 is equivalent (a)(2), which it is exactly the meaning of be -convex. So from Theorem 1.1 we have that (b) holds, that is:

For each 1 < j < n - 1, the following holds

(*) j There exist a manifold , an open neighborhood of in which is convex in the direction, and a diffeomorphism which carries each into the operator if points in are denoted by (yo, j, t).

The next step of the proof is to show that we can take Mo,j so that its image on Ω, by the diffeomorphism of (*)j, is equal to a connected component of a level set of fj. In order to prove this, consider X_i = (det(F´(x))^-1V_f,i for i = 1,..., n, so [Xi, Xj] (fk) = 0 for all i, j and k, , therefore by the Inverse Function Theorem we have that [ Xi, Xj] = 0 for all i and j. Moreover, the orbits of Xi, for 1 < i < n are equal to the corresponding Vf,i .

So for 1 < j < n - 1 (*) holds for Xj as well. From Frobenius Theorem, since the Xj 's commute, the image on Ω of M_o,j by the diffeomorphism must be orthogonal to Xj. Therefore it is a connected component of a pre-image of, and the same holds for VF,j.

Assume that there are two points p1 and p2 of M such that F(p1) = F(p2), in particular we have that f₁(p1) =f1(p2). Clearly if the characteristic curves of passing through each are the same then we have that p1 = p2, because V_f,1(f₁) ≠ 0 . So we assume that the above curves are different connected components of{x;f_j(x) = fj(p1) for j ≠ 1}.

Identifying M_0,1 with its image in (*)j, by the diffeomorphism given in (*)j , by the above it must be equal to a connected component of (M_o,1), for some real. Furthermore, since {V_F,i; 2 < i < n} are tangent to, we reduce to prove the theorem for F|M_o,1ε Φ (M_0,1, R^n-10).

We apply the argument above until we get the restriction of to a component connected of , but on this curve the restriction F of is equal to. So F must be injective there by using that V_F,nfn ≠ 0 . Concluding the proof of Theorem 0.1.

EXTENSION OF THEOREM 1.1

Now we consider an extension of Theorem 1.1 for a connected paracompact smooth oriented - dimensional manifold (M, ω), where is a globally defined non-vanishing smooth -form. We refer to (Warner 1983) for details.

Suppose that F = (f1,...,fn): M → R

ⁿ is a smooth mapping with injective derivative at any point of

M . As before such a mapping is said to belong to Φ(M) = Φ(

M,R

ⁿ) . From the hypothesis on

M we have that there is(U

_m, φ

_m) a coordinate system for

M so that U

_m is pre-compact for each

m .Take (ψ

m) m ε z to be a partition of unity associated to (U

m) . Furthermore, we can assume that the restriction of F on U

_mis a diffeomorphism over its image.

Let 1 < j < n consider the vector Vi field defined as follows, letg ε C^∞(M) , take:

It is a routine computation to show that Vi is a derivation, then it defines a vector field on M.

Now we will see that these vector fields generalize for M the vector fields considered before on the the euclidean context. In fact, let and write g = Σ_mψ_g , therefore the support of ψmg is contained on U_m . Since the restriction of F to Um is a diffeomorphism we can find a such that . . Then

Defining Det( DF(x)) to be the coefficient of in terms ofdf₁ ^...^df_n in terms of ω, we have that

As before we have that f_j , for j ≠ i , is a first integral of Vi . Also the vector fields are in involution. We consider an extension of Definition 1.1:

DEFINITION 2.1 Let F e Φ (M). We say that is -convex if there are an smooth manifold M₁, G₁Φ (M₁,M) with G₁ (M₁) and where, for j = 1,2, each Gj is a diffeomorphism over its image such that there are n -1 different indices ij,...,i_n-1 e ∈ { 1,...,n} where M₁ is V_f1,i -convex for j ∈ { 1,...,n - 1} and F₁= G₂ο F ο G₁.

From inspection on the proof of Theorem 1.1 we have:

THEOREM 2.2 Let M be a connected paracompact smooth oriented manifold and F ∈ Φ(M) . If is -convex then is injective.

Theorem 2.2 could be used to decide whether a local parametrization of a manifold, defined globally, is a global parametrization. The result below extends for arbitrary dimension Theorem 0.3 of(Santos Filho 2004):

COROLLARY 2.1 Let be be a connected paracompact smooth oriented manifold and F ∈ Φ(M) so that is Ω-convex. Then is contractible.

We observe from the proof of Theorem 1.1 that is a diffeomorphic image of y x Rn^-1 where y is a smooth curve which is diffeomorphic to an interval. Then the conclusion of the corollary follows.

REMARKS.

1) The proof that each pair of Vf,j 's is in involution can also be proved without normalizing them.That is without considering the X´_js. In fact, let and Vf,ji, with j1 ≠ j₂, then V^f,ji (fj) = 0 if for if ji ≠ for i ∈ {1,2}. Therefore [V_f,j1, V_f,j2] at any point is linear combination of V_f,ji and Vf,j2 , proving the assertion.

2) From Theorem 2.1, we can not have a vector field globally solvable on a compact manifold without boundary. Otherwise the manifold would be a diffeomorphic image of a cylinder Mo x R, where Mo is n-1 a dimensional manifold, therefore imposing a strong restriction on the topology of M.

ACKNOWLEDGMENTS

This work was partially supported by the Grant no. 2007/08231-0 from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) to José R. dos Santos Filho.

Manuscript received on November 3, 2008; accepted for publication on April 14, 2010

• BASS H, CONNELL EH AND WRIGHT D. 1982. The Jacobian conjecture: Reduction of degree and formal expansion of the inverse. Bull Am Math Soc, New Ser 7: 287-330.
• DUISTERMAAT JJ AND HÖRMANDER L. 1972. Fourier Integral Operators II. Acta Math 128: 183-269.
• MALGRANGE B. 1956. Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution. Ann Inst Fourier (Grenoble) 6: 271-355.
• SANTOS FILHO JR DOS. 2004. Injective mappings and solvable vector fields of Euclidean Spaces. Topology Appl 136: 261-274.
• WARNER F. 1983. Foundations of differentiable manifolds and Lie groups. Springer-Verlag.

Publication Dates

• Publication in this collection
  27 Aug 2010
• Date of issue
  Sept 2010

History

• Received
  03 Nov 2008
• Accepted
  14 Apr 2010