TRANSVERSALITY AND LIPSCHITZ FREDHOLM MAPS

arXiv:1508.05239v1 [math.DG] 21 Aug 2015

KAVEH EFTEKHARINASAB

Abstract. We study transversality for Lipschitz-Fredholm maps in the context of bounded Fr´echet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fr´echet spaces has the transverse stability property. We give a straightforward extension of the Smale transversality theorem by using the generalized Sard’s theorem for this category of manifolds. We also provide an answer to the well known problem concerning the existence of a submanifold structure on the preimage of a transversal submanifold.

1. Introduction

In [2] we proved a version of the classical Sard-Smale theorem for a category of generalized Fr´echet manifolds, bounded (or M Ck_{) Fr´echet manifolds, introduced in [5]. Our approach}

to the theorem’s generalization is based on the assumption that Fredholm operators need to be globally Lipschitz. A reason for this interest is that there exists an appropriate topology on L_{pE, Fq, the space of all linear globally Lipschitz maps between Fr´echet spaces E and}

F, that leads to the openness of the set of linear isomorphisms in L_{pE, Fq [2, Proposition}

2.2]. This result in turn yields the openness of the collection of Fredholm operators in

L_{pE, Fq [2, Theorem 3.2]. The other reason is that Lipschitzness is consistent with the}

notion of differentiability, bounded (or M Ck_{-) differentiability, that we apply. If E, F are}

Fr´echet spaces and if U is an open subset of E, a map f : U Ñ F is called bounded (or

M C1

-) differentiable if it is Keller-differentiable, the directional derivative dfppq belongs to

L_{pE, Fq for all pPU, and the induced map df :U Ñ}L_{pE, Fq is continuous. Thus, we can}

naturally define the index of a Fredholm map between manifolds.

We should point out that the mentioned results stems from the essential fact that under a certain condition we can endow the space L_{pE, Fq with a topological group structure.}

Also, the group of automorphisms of a Fr´echet space E, AutpEq, is open in L_{pE, Eq [3,}

Proposition 2.1]. But, in general, the group of automorphisms of a Fr´echet space does not admit a non-trivial topological group structure. Thus, without some restrictions it would be impossible to establish openness of sets of linear isomorphisms and Fredholm operators. This is a major obstruction in developing the Fredholm theory for Fr´echet spaces.

2010Mathematics Subject Classification. 58B15, 57N75, 58A99 .

A crucial step in the proof of an infinite dimensional version of Sard’s theorem is that, roughly speaking, for a Fredholm map f : M Ñ N of manifolds, at each point p P M, we may find local charts ppP U ĎM, φq and pfppq P V Ď N, ψqsuch that in the charts f has a representation of the form fpu, vq “ pu, ηpu, vqq, where η : φpUq Ñ Rn _{is some smooth}

map. This is a consequence of an inverse function theorem. One of the main significance of the category of bounded Fr´echet manifolds is the availability of an inverse function theorem in the sense of Nash and Moser [5, Theorrem 4.7]. However, the bounded differentiability is strong and in some cases the class of bounded maps can be quite small, e.g. when the identity component of L_{pE, Fqcontains only the zero map [3, Remark 2.16].}

We have argued that why we have utilized this particular category of Fr´echet manifolds. A salient example of these manifolds is the space of all smooth sections of a fiber bundle over closed or non-compact manifolds ([5, Theorem 3.34]). On the other hand, it turns out that these generalized manifolds can surpass the geometry of Fr´echet manifolds. On these manifold we are able to give a precise analytic meaning to some essential geometric objects (such as connection maps, vector fields and integral curves) [1]. Therefore, we would expect their applications to problems in global analysis.

The present work studies the differential topology of Lipschitz-Fredholm maps in the bounded Fr´echet setting. We show that the set of Lipschitz-Fredholm operators of index

l between Fr´echet spaces E and F is open in the space of linear globally Lipschitz maps endowed with the fine topology. We say that a set of maps has the transverse stability property for the fine topology if maps in a fine neighborhood of a given map have the same transversality property i.e. if f :E ÑF is a map transversal to a closed subspace F _{of F,} then any map in a fine neighborhood of f is transversal to F_{. We then prove that the set} of all Lipschitz-Fredholm maps of a fixed index between Fr´echet spaces has the transverse stability property. We also study transversality for Lipschitz-Fredholm maps on manifolds. We give a straightforward generalization of the Smale transversality theorem ([6, Theorem 3.1]) by using our generalized Sard’s theorem. Finally, we prove that if f : M Ñ N is an M Ck Lipschitz-Fredholm map of manifolds which is transversal to a finite dimensional submanifold A _{of N, then f}´1

pA_{qis a submanifold of M.}

We stress that these results can not be proved without strong restrictions. However, the basic concepts of infinite dimensional differential topology such as submanifold and transversality can be simply come over from the Banach setting.

2. Preliminaries

We shall recall the required definitions from the category ofM Ck _{manifolds briefly but in}

a self-contained way for the convenience of the reader, which also give us the opportunity to establish our notations for the rest of the paper. For more studies we refer to [2, 5, 1].

Let pF, dq be a Fr´echet space whose topology is defined by a complete translational-invariant metric d. A metric with absolutely convex balls will be called a standard metric. Every Fr´echet space admits a standard metric which defines its topology. We shall always define the topology of Fr´echet spaces with this type of metrics.

Let pE, gq and pF, dq be Fr´echet spaces and let L_{g,dpE, Fq be the set of all linear maps}

L:E ÑF such that

LippLqg,d – sup

xPEzt0_u

dpLpxq,0q

gpx,0q ă 8. The transversal-invariant metric

Dg,d:Lg,dpE, Fq ˆLg,dpE, Fq ÝÑ r0,8q, pL, Hq ÞÑLippL´Hqg,d, (2.1)

on L_{d,gpE, Fq turns it into an Abelian topological group ([2, Remark 2.1]). A map ϕ P} L_{g,dpE, Fq is called Lipschitz-Fredholm operator if its kernel has finite dimension and its}

image is closed and has finite co-dimension. The index of ϕ, Indϕ, is defined by Indϕ “ dim kerϕ´codim Imgϕ.We denote byLF_{pE, Fqthe set of all Lipschitz-Fredholm operators,}

and by LF_{lpE, Fqthe subset of} LF_{pE, Fq consisting of those operators of index l.}

Proposition 2.1. [2, Proposition 2.2] The set of linear isomorphisms from E into F,

IsopE, Fq, is open in L_{d,gpE, Fq with respect to the topology induced by the Metric (2.1).}

Theorem 2.1 ([2], Theorem 3.2). LF_{pE, Fq}is open in L_{g,dpE, Fq} with respect to the topol-ogy defined by the Metric (2.1). Furthermore, the function T Ñ IndT is continuous on

LF_{pE, Fq}, hence constant on connected components of LF_{pE, Fq}.

A subset G of a Fr´echet space F is called topologically complemented (or it splits inF), if F is homeomorphic to the topological direct sumG‘H, where H is a subspace ofF. We call H a topological complement of G inF.

Theorem 2.2 ([5], Theorem 3.14). Let E be a Fr´echet space. Then

(1) Every finite-dimensional subspace of E is closed.

(2) Every closed subspace G Ă E with codimpGq “ dimpE{Gq ă 8 is topologically complemented in E.

(3) Every finite-dimensional subspace of E is topologically complemented.

Let E, F be Fr´echet spaces, U an open subset of E, and P : U ÑF a continuous map. Let CLpE, Fq be the space of all continuous linear maps from E to F topologized by the compact-open topology. We say P is differentiable at the point p P U if the directional derivative dPppq exists in all directions h P E. If P is differentiable at all points p P U, if dPppq : U Ñ CLpE, Fq is continuous for all p P U and if the induced map P1 : U ˆE Ñ

F, pu, hq ÞÑ dPpuqh is continuous in the product topology, then we say that P is Keller-differentiable. We define Ppk`1_q

:UˆEk`1

ÑF in the obvious inductive fashion.

If P is Keller-differentiable, dPppq P L_{d,gpE, Fq for all p P U, and the induced map}

dPppq:U ÑL_{d,gpE, Fq is continuous, then P is called bounded differentiable. We say P is}

M C0

and writeP0

“P if it is continuous. We say P is an M C1

and write Pp1_q

“P1 if it is bounded differentiable. LetL_{d,gpE, Fq}0be the connected component ofL_d,gpE, Fqcontaining

the zero map. IfP is bounded differentiable and ifV ĎU is a connected open neighborhood ofx0 PU, thenP1pVqis connected and hence contained in the connected componentP1px0q`

L_{d,gpE, Fq}0 of P1px0q in L_d,gpE, Fq. Thus, P1 |_V ´P1px0q: V ÑL_d,gpE, Fq0 is again a map

between subsets of Fr´echet spaces. This enables a recursive definition: If P is M C1

and V

can be chosen for eachx0 P U such thatP1 |V ´P1px0q:V ÑLd,gpE, Fq0isM Ck´ 1

, thenP is called anM Ck_{-map. We make a piecewise definition ofP}pkq_byPpkq _|

V–pP1 |V ´P1px0qqpk´

1_q

for x0 and V as before. The map P is M C8 (or smooth) if it is M Ck for all k P N0. We

shall denote the derivative ofP atpby DPppq. Note thatM Ck_{-differentiability implies the}

usual Ck_{-differentiability for maps of finite dimensional manifolds.}

Within this framework we can define M Ck _{Fr´echet manifolds, M C}k_{-maps of manifolds}

and tangent bundle over M Ck _{manifolds in obvious fashion way. We assume that manifolds}

are connected and second countable.

Letf :M ÑN pk ŕ1qbe anM Ck_{-map of manifolds. We denote byT}

xf :TxM ÑTfpxqN

the tangent map off atxPM from the tangent spaceTxM to the tangent spaceTfpxqN. We

say thatfis an immersion (resp. submersion) provided Txf is injective (resp. surjective) and

the range ImgpTxfq (resp. the kernel kerpTxfq) splits inTfpxqN (resp. TxM) for any xPM.

An injective immersion f :M ÑN which gives an isomorphism onto a submanifold of N is called an embedding. A point x P M is called a regular point if Dfpxq : TxM ÝÑ TfpxqN

is surjective. The corresponding value fpxq is a regular value. Points and values other than regular are called critical points and values, respectively.

Let M and N be M Ck _{manifolds, k ŕ 1. A Lipschitz-Fredholm map is an M C}1

-map

f :M ÑN such that for each xPM the derivative Dfpxq:TxM ÝÑTfpxqN is a

Lipschitz-Fredholm operator. The index of f, denoted by Indf, is defined to be the index of Dfpxq for some x. Since f is M Ck _{and M is connected in the light of Theorem 2.1 the definition}

3. Transversality and openness

Let F1 be a linear closed subspace of a Fr´echet space F that splits in F. Given M Ck

manifold M modelled on F, a subset M1 of M is a submanifold of M modelled on F1

provided there isM Ck_-atlastpU

i, φiquiPI onM that induces an atlas onM1, i.e. for anyiPI

there are open subsets Vi, Wi of F, F1 such that φ_ipU_iq “ V_i‘W_i and φ_ipU_iXM1q “V_i‘ t0u

is open inF1. We say thatM1 is a submanifold of Banach type if F1 is a Banach space, and

a submanifold of finite type if F1 “Rn for some n PN.

Let CpE,R`<sub>q be the set of all continuous functions from E into</sub> R`_{, h P} L_{g,dpE, Fq and}

εPCpE,R`_{q. A map f P}L_{g,dpE, Fqis called a ε-approximation to hif dpfpxq, hpxqq őεpxq}

for all x P E, we write dpf, hq ă ε for short. If we take the ε-approximation to h to be a neighborhood ofh in the set L_{g,dpE, Fq, then we obtain a topology. This topology is called}

the fine topology and we denote the resulting space by L0

f inepE, Fq.

Let M and N beM Ck _{manifolds modelled on Fr´echet spaces E and F, respectively. Let}

M Ck_{pM, Nq,1 ő k ő 8, be the set of M C}k_{-maps from M into N. Two maps f, h P}

M CkpM, Nq are said to be equivalent atxPM if Txkf “Txkh, where Tk is thek-th tangent

map. We define the k-jet of f atx, jxkf, to be the equivalence class of f. Let dk be a fiber

metric on the tangent space Tk

xM that induces a Fr´echet topology which is isomorphic toE.

We describe the fine topology of order k on M Ck_{pM, Nq as follows. Let ϕ P M C}k_{pM, Nq}

and Ω–_{tViuiPI be a locally finite cover of M. Letǫi :Vi Ñ}R` _{be continuous for alli PI.} Then, the sets

Θpϕ, Vi, ǫiq–tφ PM CkpM, Nq |dkpjxkφ, jxkϕq ăǫipxq, xPViu

constitute a basis for fine open neighborhoods of ϕ. In this case we say that φ in a fine neighborhood of ϕ is an M Ck _{fine approximation to ϕ.}

Lemma 3.1. The fine topology is finer than the topology induced by the Metric (2.1). Proof. We must show that if N_{pf, δq is a δ-neighborhood of f, then we can find ǫą0 such}

that if Dg,dpf, hq ă ǫ, then hPNpf, δq. Given a map hPLg,dpE, Fq, let

ǫ–_{mint1, inf}

xPEzt0_u

δpxq

gphpxq,0qu.

Now suppose Dg,dpf, hq ăǫ, then we can easily see that dpf, hq ă δ and hence h PNpf, δq.

Remark 3.1. We know thatpProposition 2.1q IsopE, Fq is open inL_{g,dpE, Fq}endowed with the topology induced by the metric (2.1). By the preceding lemma the fine topology is finer

than the metric topology, thereby, IsopE, Fq is open in L0

Definition 3.1. Let f :E ÑF be a Lipschitz-Fredholm operator of Fr´echet spaces. We say that f is transversal to a closed subset F0 ĎF and write f&_F0 if

(1) Imgpfq `F0 “F, and

(2) either F0 splits in F or f´ 1

pF0q splits in E.

The following result characterizes the transversality of Lipschitz-Fredholm operators.

Proposition 3.1. Letϕ PLF_{lpE, Fq}. SupposeF0 ĎF is a closed subset such thatImgpϕq` F0 “ F. Then ϕ&_F0 if and only if there are closed subsets F1 Ď F and E0 Ď E with F “F0‘F1 and E “E0‘ pE1 –ϕ´

1

pF1qq such that ϕ1 –ϕ|_E1 PIsopE1, F1q.

Proof. Assume that such a closed subset F0 is given and ϕ&F0. pImgpϕq XF0q splits in

F0 because m “ dimpF0{Imgpϕqq ő dimpF{Imgpϕqq ă 8 and hence by Theorem 2.2(2)

there exists a space F _{Ď F0 of dimension m such that F0 “ pImgpϕq X F0q ‘} F_{. Since} Imgpϕq XF_{ĎImgpϕq XF}0 it follows that Imgpϕq XF“ t0u. Also, Imgpϕq `F“ pImgpϕq `

pImgpϕqXF0qq`F“Imgpϕq`F0 “F. Thus, Imgpϕq‘F“F, therefore, codim Imgpϕq “ m

and dim kerpϕq “ l `m. Moreover, there exists a closed subset E _{Ď E such that E “} kerpϕq ‘E_{. The operator Φ} – _ϕ|E P LpE,Imgpϕqq is injective onto Imgpϕq, hence, by virtue of open mapping theorem is a homeomorphism and therefore ΦPIsopE_{,Imgpϕqq. Let} E0 –Φ´

1

pImgpϕq XF0q ĎE, then E0 “ϕ´ 1

pImgpϕq XF0q “kerpϕq ‘E0.

E0 is complemented in E0 so there is a continuous projection Pr1 of E0 onto E0 (see

Theorem 2.2(4)). IfE0 is complemented in E, then there exists a continuous projection Pr2

of E ontoE0. Thus, Pr1˝Pr2 is a continuous projection fromE toE0 and its restriction to

E _{is a again continuous projection onto} E0, thereby, E0 is complemented in E. This means

there is a closed subset E1 ĎE (which is also closed in E) such that E“E1 ‘E0.

By the same argument we have, ifF0 is complemented inF, thenpImgpϕq XF0qis

comple-mented in ImgpϕqbecausepImgpϕqXF0qis complemented inF0. This means there is a closed

subspace F1 Ď Imgpϕq (which is also closed in F) such that Imgpϕq “ F1 ‘ pImgpϕq XF0q.

Therefore, we haveE “kerpϕq‘E0‘E1 “E0‘E1 andF “ pImgpϕqXF0q‘F‘F1 “F0‘F1

and ϕ1 “Φ|_E1 PIsopE1, F1q. Moreover,E1 “ϕ

´1

1 pF1q. The converse is obvious.

Proposition 3.2. LF_{lpE, Fq} is open in L0

f inepE, Fq.

Proof. Let ϕ P LF_{lpE, Fq. We show that there exists ε ą 0 such that any φ P} L_{g,dpE, Fq}

which is ε-approximation to ϕ is a Lipschitz-Fredholm operator of indexl.

kerpLq ‘kerpϕq “E and ImgpLq ‘Imgpϕq “F. Choose εP p0,1{2LippK´1

qqsmall enough and suppose that φ,L _P L_{pE, Fq are ε-approximation to ϕ and the dimension of the image}

of L _{is finite. Then} K_“ L_{`φ satisfies dpKpxq,}K_{pxqq ă1{LippK}´1

q, for all x P E, thus K is an isomorphism (see Remark 3.1) and hence φP LF_{pE, Fq and Indpφq “ 0.}

Now suppose l ą0, define the linear globally Lipschitz operators ϕl, φl : E Ñ F ˆRl by

ϕlpxq–pϕpxq,0qand φlpxq –pφpxq,0q. Then ϕl is a Lipschitz-Fredholm operator of index

0. By the above argument φl is a Lipschitz-Fredholm operator of index 0 and hence φ is a

Lipschitz-Fredholm operator of index l. Likewise, the case lă0 can be proved.

Theorem 3.1. Let ϕP LF_{lpE, Fq}, and suppose thatF0 ĎF is closed andϕ&_{F0. Then any}

φP L_{g,dpE, Fq in a fine neighborhood of ϕ is transversal to F}0.

Proof. By Proposition 3.1 there exist closed subsetsE0 ĎE,F1 ĎF andE1 –_ϕ´1_pF1qsuch

that F “F0‘F1, E “E0‘E1 andϕ1 –ϕ|_E1 PIsopE1, F1q. There is a continuous function

δpxq such that every linear globally Lipschitz map ψ : E1 Ñ F1 which is δ-approximation

to ϕ1 is an isomorphism (see Remark 3.1). Let π : F Ñ F1 be the projection given by πpf0`f1q “f1, and let κ“IdF ´π. It is immediate that π is linear and globally Lipschitz

and Imgpκq “F0. Chooseε P p0, δ{Lippπqq small enough, in view of Proposition 3.2, we can assume that each φ PL_{pE, Fqwhich is ε-approximation toϕ belongs to} LF_{lpE, Fq.}

Now we show that each such φ is transversal toF0. Let Φ–pπ˝ϕq|_E1 PLpE1, F1q. Then

dpΦ, ϕ1q ő Lippπqε ă δ and so Φ P IsopE1, F1q (see Remark 3.1). Thus, we only need to

prove F “ Imgpφq ` F0. Let f P F “ F0 ‘F1 so f “ f0 `f1, where f_i P F_ipi “ 0,1q.

We have Φ´1

pf1q “ e1 P E1 Ď E, x “ φpe1q P Imgpφq, and y “ f0 ´κpxq P F0. Whence,

x`y“πpxq `κpxq `f0´κpxq “f0`Φpe1q “f0`f1 “f, therefore F “Imgpφq `F0.

Now we prove the transversality theorem forM Ck_{-Lipschitz-Fredholm maps. It is indeed}

a consequence of the Sard’s theorem for these maps [2, Theorem 4.3]. A careful reading of the proof of the theorem shows that the minor assumption of endowing manifolds with compat-ible metrics is superfluous and the theorem remains valid for manifolds without compatcompat-ible metrics. Thus, the statement of the theorem is as follows:

Theorem 3.2 (Sard’s Theorem). Let M and N be M Ck _{manifolds, k ŕ 1. If f :M ÑN}

is an M Ck-Lipschitz-Fredholm map with k ąmaxtIndf,0u. Then, the set of regular values

of f is residual in N.

Definition 3.2. Letf :M ÑN be a Lipschitz-Fredholm map and let ı:A ã_{ÑN be anM C}1 embedding of a finite dimensional manifold A. We say that f is transversal to ı and write

The following theorem is the analogous of the Smale transversality [6, Theorem 3.1]. Its proof is just a slight modification of the argument of Smale.

Theorem 3.3. Let M and N be M Ck _{manifolds modelled on spaces pF, dq and pE, gq,}

respectively. Let f : M Ñ N be an M Ck_{-Lipschitz-Fredholm map and let ı : A ãÑ N}

be an M C1-embedding of a finite dimension manifold A with k ą maxtIndf `dimA_,0u.

Then there exists an M C1

fine approximation g of ı such that g is embedding and f&_g.

Furthermore, Suppose S is a closed subset of A and f&_{ıpSq, then g can be chosen so that}

ı“g on S.

Proof. Since manifolds are second countable we only need to work in local coordinates. Assume that y P A _{and n “ dimıp}A_{q. Since ıp}A_{q is an embedded submanifold of finite}

type of N, we may find an open neighborhood U Ă Rn _{about y, a chart about ıpyq and a}

splitting E “Rn_‘E1 such that ıpyq “ıpy,0qin the neighborhoods. Letπ2 :E ÑE1 be the

projection onto E1. LetV ĂU be a neighborhood ofy, and ha smooth real valued function

which is 1 onV and 0 outside of U. Since π1 ˝f is locally Fredholm-Lipschitz it follows by

Sard’s Theorem (Theorem 3.2) that there is a regular value z for π1˝f which is close to 0.

Now define

gpyq “hpyqpz, yq ` r1´hpyqsıpyq.

It is immediate thatf&_{gonV, and forz sufficiently close to 0,g isM C}1

fine approximation

toı. The second statement follows by our definition ofg.

4. Transversal submanifolds

We will need the following inverse function theorem.

Theorem 4.1 ([5], Theorem 4.7. Inverse Function Theorem for M Ck-maps). Let pE, gq be a Fr´echet space with standard metric g. Let U ĂE be open, x0 PU and f :U ĂE ÑE an

M Ck_{-map, k ě1. If f}1_px

0q PAutpEq, then there exists an open neighborhood V ĎU of x0 such that fpVq is open in E and f|V :V ÑfpVq is an M Ck- diffeomorphism.

To avoid some technical complications we consider only manifolds without boundary in the sequel.

Theorem 4.2. Let M and N be M Ck manifolds modelled on spaces pF, dq and pE, gq, respectively. Suppose that f : M Ñ N is an M Ck_{-Lipschitz-Fredholm map of index l. Let} A be a submanifold of N with dimension m and let ı : A ã_{Ñ N be the inclusion. If f is}

transversal toA, thenf´1

pA_q is a submanifold ofM of dimensionl`m. For allxP f´1

pA_q

we have Txpf´

1

pA_{qq “ pTxfq}´1

Proof. Iff´1

pA_{q “ Hthe theorem is clearly valid so let f}´1

pA_{q ‰ H. Let pψ, Uq be a chart}

atfpx0q PA_{inN with the submanifold property for}A_{. LetU1, U2 be open subsets ofE,}Rm such that ψpUq “ U1 ‘U2, ψpU XAq “ U1 ‘ t0u, and ψpfpx0qq “ p0,0q. Let pV, ϕq be a

chart at x0 inM such that ϕpx0q “0, ϕ:V ÑϕpVq ĂF and fpVq Ă U. Let

f –_{ψ ˝f ˝ϕ}´1 _{:ϕpVq Ñ ψpUq}

be the local representative of f. Then fp0q “ p0,0q and by hypothesis f is a Lipschitz-Fredholm map, in particular, Dfp0q P LF_{lpF, Eq. The tangent map Tfpx0qı : Tfpx0q}A _Ñ

Tfpx0qN is injective with closed split image. Hence Tfpx0qA can be identified with a closed split subspace of Tfpx0qN. Thus Dfpx0q is transversal to Tfpx0qA. Therefore, keeping in the mind the definition of the differential in terms of tangent maps, Dfp0q is transversal to

T ψpTfpx0qAq “ U1 ‘ t0u “: E1. Then, by virtue of Proposition 3.1 there are closed subsets

F1 Ă F, E0 Ă E such that F “ F1 ‘ pF0 – Dfp0qpE0qq, E “ E1 ‘E0, ∆ – Dfp0q |_F0P IsopF0, E0q and ∆1 –Dfp0q |_F1P IsopF1, E1q. Moreover, dimF0 “m`l.

Consider the projection π : F Ñ F1 given by πpf0 `f1q “ f1. Since F1 and F0 are

closed and complementary it follows that obviously the map κ “ IdF ´π is the unique

projection with Imgpκq “ F0 and kerpκq “ F1 . Let π1 : E Ñ E0 be the projection given

by π1pe0 `e1q “ e0. Then, Π – _∆´1_{˝π1 ˝D}fp0q is a projection with ImgpΠq “ F0 and

F1 ĎkerpΠq. Since F “F0‘F1, it follows that F1 “kerpΠq and therefore Π“κ. Now define the map H : ϕpVq Ñ F of class M Ck _{by Hpxq “ πpxq `∆}´1

˝π1 ˝fpxq.

We obtain that Hp0q “ 0 and DHp0q “ π ` ∆´1

˝π1 ˝ Dfp0q “ π `κ “ Id_F. If we

choose V small enough, then by Theorem 4.1 H is an M Ck_{-diffeomorphism onto an open}

neighborhood U _{ĎψpUq of ψpfpx}0q “ p0,0q. LetΦ“H˝ϕ´ 1

, thenpΦ, F0qis a chart for x0

onV with the submanifold property. Because we have

xPf´1pA_{q ôψpfpxqq PU}1‘ t0u ôfpϕpxqq PU1‘ t0u ô Hpϕpxqq PF0.

Let pP A_{, γ :}R_{ÑM a smooth curve sending zero top, and j}1

pγ the 1-jet of γ atp.

j1

pγ PTpAôj

1

ϕppqpϕ˝γq “T ϕpj 1

pγq PϕpVq ˆF, ϕ˝γ ĂϕpVq

ôTfpjϕ1ppqpϕ˝γqq PψpUq ˆE

ôfpϕ˝γq “ψpf˝γq ĂψpUq ôd

dtψpf˝γq |t“0“v “

d dtψprψ

´1

pψpfpxqq `tvqsq, ψpf˝γq ĂψpUq

ôjf1ppqpf˝γq “j 1

prψ´

1

pψpfppqq `tvqs P TfppqA

ôjp1γ P pTpfq´

1

pTfppqAq

If manifolds have nonempty boundary we just need to modify the proof by extending the considered maps.

Corollary 4.1. Let f : M Ñ N be an M Ck_{-Lipschitz-Fredholm map of index l. If y is a}

regular value of f, then the level set f´1

pyq is a submanifold of dimension l and its tangent space at x is kerTxf.

Proof. The settyu is transversal to f so the result follows from the theorem.

Corollary 4.2. Let f :M ˆN ÑO be a smooth Lipschitz-Fredholm map of manifolds, we write fx – fp¨, xq, and let A be a closed finite dimension submanifold of O. Assume that

f&A and for all pm, nq P f´1

n pAq the composition pTmM

Dfnpmq

ÝÝÝÝÝÑTfnpmqO

Q

ÝÑTfnpmqO{TnSq

is Lipschitz-Fredholm. Then there is a residual set of n in O for which the map fn :M ÑO

is transversal to A.

Proof. By hypothesis the kernel of Q˝Dfpxq is complemented for all x P f´1

pA_{q. By the}

preceding theoremB –_f´1

pA_{qis a Fr´echet submanifold. The mapf |Bis smooth}

Lipschitz-Fredholm, therefore by Sard’s theorem there is a residual set of regular values of it inO. If

nP N is a regular value of f |B, then fn is transversal to A.

References

[1] K. Eftekharinasab, Geometry of bounded manifolds, Rocky Mountain J. Math., to appear

[2] K. Eftekharinasab, Sard’s theorem for mappings between Fr´echet manifolds, Ukrainian Math. J. 62

(2010), 1634-1641.

[3] H. Gl¨ockner, Implicit functions from topological vector spaces in the presence of metric estimates, preprint, Arxiv:math/6612673, 2006.

[4] G. K¨oethe, Topological vector spaces I, Springer-Verlag, New York Inc., 1969.

[5] O. M¨uller, A metric approach to Fr´echet geometry, Journal of Geometry and physics 58 _(2008),

1477-1500.

[6] Smale S. An infinite dimensional version of Sard’s theorem,Amr.J.Math.4_{(1965) 861-866.}

Topology dept., Institute of Mathematics of NAS of Ukraine, Tereshchenkivska st. 3, Kyiv, 01601 Ukraine