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Differential Geometry and its Applications 1 (1991) 391–401

arXiv:math/9202208v1 [math.DG] 1 Feb 1992Differential Geometry and its Applications 1 (1991) 391–401North-HollandTHE ACTION OF THE DIFFEOMORPHISM GROUPON THE SPACE OF IMMERSIONSVicente CerveraFrancisca Mascar´oPeter W. MichorDepartamento de Geometr´ıa y Topolog´ıaFacultad de Matem´aticasUniversidad de ValenciaInstitut f¨ur Mathematik, Universit¨at Wien,Strudlhofgasse 4, A-1090 Wien, Austria.1989Abstract.We study the action of the diffeomorphism group Diff(M) on the space of properimmersions Immprop(M, N) by composition from the right.We show that smooth transversalslices exist through each orbit, that the quotient space is Hausdorffand is stratified into smoothmanifolds, one for each conjugacy class of isotropy groups.Table of contentsIntroduction1. Regular orbits2.

Some orbit spaces are Hausdorff3. Singular orbits1991 Mathematics Subject Classification.

58D05, 58D10.Key words and phrases. immersions, diffeomorphisms.This paper was prepared during a stay of the third author in Valencia, by a grant given by Coseller´ıa deCultura, Educaci´on y Ciencia, Generalidad Valenciana.

The two first two authors were partially supported bythe CICYT grant n. PS87-0115-G03-01.Typeset by AMS-TEX1

2CERVERA, MASCAR ´O, MICHORIntroductionLet M and N be smooth finite dimensional manifolds, connected and second countablewithout boundary such that dim M ≤dim N. Let Imm(M, N) be the set of all immersionsfrom M into N. It is an open subset of the smooth manifold C∞(M, N), see our main reference[Michor, 1980c], so it is itself a smooth manifold. We also consider the smooth Lie group Diff(M)of all diffeomorphisms of M. We have the canonical right action of Diff(M) on Imm(M, N) bycomposition.The space Emb(M, N) of embeddings from M into N is an open submanifold of Imm(M, N)which is stable under the right action of the diffeomorphism group.

Then Emb(M, N) is thetotal space of a smooth principal fiber bundle with structure group the diffeomorphism group;the base is called B(M, N), it is a Hausdorffsmooth manifold modeled on nuclear (LF)-spaces.It can be thought of as the ”nonlinear Grassmannian” of all submanifolds of N which are oftype M. This result is based on an idea implicitly contained in [Weinstein, 1971], it was fullyproved by [Binz-Fischer, 1981] for compact M and for general M by [Michor, 1980b]. Theclearest presentation is in [Michor, 1980c, section 13].

If we take a Hilbert space H insteadof N, then B(M, H) is the classifying space for Diff(M) if M is compact, and the classifyingbundle Emb(M, H) carries also a universal connection. This is shown in [Michor, 1988].The purpose of this note is to present a generalization of this result to the space of immersions.It fails in general, since the action of the diffeomorphism group is not free.

Also we were notable to show that the orbit space Imm(M, N)/ Diff(M) is Hausdorff. Let Immprop(M, N) be thespace of all proper immersions.

Then Immprop(M, N)/ Diff(M) turns out to be Hausdorff, andthe space of those immersions, on which the diffeomorphism group acts free, is open and is thetotal space of a smooth principal bundle with structure group Diff(M) and a smooth manifoldas base space. For the immersions on which Diff(M) does not act free we give a slice theoremwhich is explicit enough to describe the stratification of the orbit space in detail.

The resultsare new and interesting even in the special case of the loop space C∞(S1, N) ⊃Imm(S1, N).The main reference for manifolds of mappings is [Michor, 1980c]. But the differential calculusused there is a little old fashioned now, so it should be supplemented by the convenient settingfor differential calculus presented in [Fr¨olicher-Kriegl, 1988].If we assume that M and N are real analytic manifolds with M compact, then all infinitedimensional spaces become real analytic manifolds and all results of this paper remain true, byapplying the setting of [Kriegl-Michor, 1990].1.

Regular orbits1.1.Setup. Let M and N be smooth finite dimensional manifolds, connected and secondcountable without boundary, and suppose that dim M ≤dim N. Let Imm(M, N) be the man-ifold of all immersions from M into N and let Immprop(M, N) be the open submanifold of allproper immersions.Fix an immersion i.

We will now describe some data for i which we will use throughout thepaper. If we need these data for several immersions, we will distinguish them by appropriatesuperscripts.First there are sets Wα ⊂W α ⊂Uα ⊂M such that (Wα) is an open cover of M, W α iscompact, and Uα is an open locally finite cover of M, each Wα and Uα is connected, and suchthat i|Uα : Uα →N is an embedding for each α.

THE SPACE OF IMMERSIONS3Let g be a fixed Riemannian metric on N and let expN be its exponential mapping. Thenlet p : N(i) →M be the normal bundle of i, defined in the following way: For x ∈M letN(i)x := (Txi(TxM))⊥⊂Ti(x)N be the g-orthogonal complement in Ti(x)N. ThenN(i)¯i−−−−→T NpyyπNM−−−−→iNis a vector bundle homomorphism over i, which is fiberwise injective.Now let U i = U be an open neighborhood of the zero section which is so small that(expN ◦¯i)|(U|Uα) : U|Uα →N is a diffeomorphism onto its image which describes a tubu-lar neighborhood of the submanifold i(Uα) for each α. Letτ = τ i := (expN ◦¯i)|U : N(i) ⊃U →N.It will serve us as a substitute for a tubular neighborhood of i(M).1.2.

Definition. An immersion i ∈Imm(M, N) is called free if Diff(M) acts freely on it, i.e.if i ◦f = i for f ∈Diff(M) implies f = IdM.

Let Immfree(M, N) denote the set of all freeimmersions.1.3. Lemma.

Let i ∈Imm(M, N) and let f ∈Diff(M) have a fixed point x0 ∈M and satisfyi ◦f = i. Then f = IdM.Proof.

We consider the sets (Uα) for the immersion i of 1.1. Let us investigate f(Uα) ∩Uα.

Ifthere is an x ∈Uα with y = f(x) ∈Uα, we have (i|Uα)(x) = ((i ◦f)|Uα)(x) = (i|Uα)(f(x)) =(i|Uα)(y). Since i|Uα is injective we have x = y, andf(Uα) ∩Uα = {x ∈Uα : f(x) = x}.Thus f(Uα) ∩Uα is closed in Uα.

Since it is also open and since Uα is connected, we havef(Uα) ∩Uα = ∅or = Uα.Now we consider the set {x ∈M : f(x) = x}. We have just shown that it is open in M.Since it is also closed and contains the fixed point x0, it coincides with M.□1.4.

Lemma. If for an immersion i ∈Imm(M, N) there is a point in i(M) with only onepreimage, then i is a free immersion.Proof.

Let x0 ∈M be such that i(x0) has only one preimage. If i ◦f = i for f ∈Diff(M) thenf(x0) = x0 and f = IdM by lemma 1.3.□Note that there are free immersions without a point in i(M) with only one preimage: Considera figure eight which consists of two touching circles.

Now we may map the circle to the figureeight by going first three times around the upper circle, then twice around the lower one. Thisimmersion S1 →R2 is free.

4CERVERA, MASCAR ´O, MICHOR1.5. Theorem.

Let i be a free immersion M →N. Then there is an open neighborhood W(i)in Imm(M, N) which is saturated for the Diff(M)-action and which splits smoothly asW(i) = Q(i) × Diff(M).Here Q(i) is a smooth splitting submanifold of Imm(M, N), diffeomorphic to an open neighbor-hood of 0 in C∞(N(i)).

In particular the space Immfree(M, N) is open in C∞(M, N).Let π : Imm(M, N) →Imm(M, N)/ Diff(M) = B(M, N) be the projection onto the orbitspace, which we equip with the quotient topology. Then π|Q(i) : Q(i) →π(Q(i)) is bijectiveonto an open subset of the quotient.

If i runs through Immfree,prop(M, N) of all free and properimmersions these mappings define a smooth atlas for the quotient space, so that(Immfree,prop(M, N), π, Immfree,prop(M, N)/ Diff(M), Diff(M))is a smooth principal fiber bundle with structure group Diff(M).The restriction to proper immersions is necessary because we are only able to show thatImmprop(M, N)/ Diff(M) is Hausdorffin section 2 below.Proof. We consider the setup 1.1 for the free immersion i. LetU(i) := {j ∈Imm(M, N) : j(Wiα) ⊆τi(U i|U iα) for all α, j ∼i},where j ∼i means that j = i offsome compact set in M. Then by [Michor, 1980c, section 4]the set U(i) is an open neighborhood of i in Imm(M, N).

For each j ∈U(i) we defineϕi(j) : M →U i ⊆N(i),ϕi(j)(x) := (τ i|(U i|U iα))−1(j(x)) if x ∈W iα.Then ϕi : U(i) →C∞(M, N(i)) is a mapping which is bijective onto the open setV(i) := {h ∈C∞(M, N(i)) : h(Wiα) ⊆U i|U iα for all α, h ∼0}in C∞(M, N(i)). Its inverse is given by the smooth mapping τ i∗: h 7→τ i ◦h, see [Michor, 1980c,10.14].

We claim that ϕi is itself a smooth mapping: recall the fixed Riemannian metric g onN; τ i is a local diffeomorphism U i →N, so we choose the exponential mapping with respectto (τ i)∗g on U i and that with respect to g on N; then in the canonical chart of C∞(M, U i)centered at 0 and of C∞(M, N) centered at i as described in [Michor, 1980c, 10.4], the mappingϕi is just the identity.We have τ i∗(h ◦f) = τ i∗(h) ◦f for those f ∈Diff(M) which are near enough to the identityso that h ◦f ∈V(i). We consider now the open set{h ◦f : h ∈V(i), f ∈Diff(M)} ⊆C∞((M, U i)).Obviously we have a smooth mapping from it into C∞c (U i) × Diff(M) given by h 7→(h ◦(p ◦h)−1, p ◦h), where C∞c (U i) is the space of sections with compact support of U i →M.

So if welet Q(i) := τ i∗(C∞c (U i) ∩V(i)) ⊂Imm(M, N) we haveW(i) := U(i) ◦Diff(M) ∼= Q(i) × Diff(M) ∼= (C∞c (U i) ∩V(i)) × Diff(M),

THE SPACE OF IMMERSIONS5since the action of Diff(M) on i is free. Consequently Diff(M) acts freely on each immersionin W(i), so Immfree(M, N) is open in C∞(M, N).

Furthermoreπ|Q(i) : Q(i) →Immfree(M, N)/ Diff(M)is bijective onto an open set in the quotient.We now consider ϕi ◦(π|Q(i))−1 : π(Q(i)) →C∞(U i) as a chart for the quotient space. Inorder to investigate the chart change let j ∈Immfree(M, N) be such that π(Q(i))∩π(Q(j)) ̸= ∅.Then there is an immersion h ∈W(i) ∩Q(j), so there exists a unique f0 ∈Diff(M) (given byf0 = p ◦ϕi(h)) such that h ◦f −10∈Q(i).

If we consider j ◦f −10instead of j and call it again j,we have Q(i) ∩Q(j) ̸= ∅and consequently U(i) ∩U(j) ̸= ∅. Then the chart change is given asfollows:ϕi ◦(π|Q(i))−1 ◦π ◦(τ j)∗: C∞c (U j) →C∞c (U i)s 7→τ j ◦s 7→ϕi(τ j ◦s) ◦(pi ◦ϕi(τ j ◦s))−1.This is of the form s 7→β ◦s for a locally defined diffeomorphism β : N(j) →N(i) which is notfiber respecting, followed by h 7→h ◦(pi ◦h)−1.

Both composants are smooth by the generalproperties of manifolds of mappings. So the chart change is smooth.We have to show that the quotient space Immprop,free(M, N)/ Diff(M) is Hausdorff.

Thiswill be done in section 2 below.□2. Some orbit spaces are Hausdorff2.1.

Theorem. The orbit space Immprop(M, N)/ Diff(M) of the space of all proper immersionsunder the action of the diffeomorphism group is Hausdorffin the quotient topology.The proof will occupy the rest of this section.

We want to point out that we believe that thewhole orbit space Imm(M, N)/ Diff(M) is Hausdorff, but that we were unable to prove this.2.2. Lemma.

Let i and j ∈Immprop(M, N) with i(M) ̸= j(M) in N. Then their projectionsπ(i) and π(j) are different and can be separated by open subsets in Immprop(M, N)/ Diff(M).Proof. We suppose that i(M) ⊈j(M) = j(M) (since proper immersions have closed images).Let y0 ∈i(M) \ j(M), then we choose open neighborhoods V of y0 in N and W of j(M) in Nsuch that V ∩W = ∅.

We consider the setsV := {k ∈Immprop(M, N) : k(M) ∩V ̸= ∅}andW := {k ∈Immprop(M, N) : k(M) ⊆W}.Then V and W are Diff(M)-saturated disjoint open neighborhoods of i and j, respectively, soπ(V) and π(W) separate π(i) and π(j) in Immprop(M, N)/ Diff(M).□2.3. For a proper immersion i : M →N and x ∈i(M) let δ(x) ∈N be the number of points ini−1(x).

Then δ : i(M) →N is a mapping.

6CERVERA, MASCAR ´O, MICHORLemma. The mapping δ : i(M) →N is upper semicontinuous, i.e.

{x ∈i(M) : δ(x) ≤k} isopen in i(M) for each k.Proof. Let x ∈i(M) with δ(x) = k and let i−1(x) = {y1, .

. .

, yk}). Then there are pairwisedisjoint open neighborhoods Wn of yn in M such that i|Wn is an embedding for each n. Theset M \ (Sn Wn) is closed in M, and since i is proper the set i(M \ (Sn Wn)) is also closed ini(M) and does not contain x.

So there is an open neighborhood U of x in i(M) which does notmeet i(M \ (Sn Wn)). Then obviously δ(z) ≤k for all z ∈U.□2.4.

We consider two proper immersions i1 and i2 ∈Immprop(M, N) such that i1(M) =i2(M) =: L ⊆N. Then we have mappings δ1, δ2 : L →N as in 2.3.2.5.Lemma.

In the situation of 2.4, if δ1 ̸= δ2 then the projections π(i1) and π(i2) aredifferent and can be separated by disjoint open neighborhoods in Immprop(M, N)/ Diff(M).Proof. Let us suppose that m1 = δ1(y0) ̸= δ2(y0) = m2.

There is a small connected openneighborhood V of y0 in N such that i−11 (V ) has m1 connected components and i−12 (V )has m2 connected components. This assertions describe Whitney C0-open neighborhoods inImmprop(M, N) of i1 and i2 which are closed under the action of Diff(M), respectively.

Obvi-ously these two neighborhoods are disjoint.□2.6. We assume now for the rest of this section that we are given two immersions i1 andi2 ∈Immprop(M, N) with i1(M) = i2(M) =: L such that the functions from 2.4 are equal:δ1 = δ2 =: δ.Let (Lβ)β∈B be the partition of L consisting of all pathwise connected components of levelsets {x ∈L : δ(x) = c}, c some constant.Let B0 denote the set of all β ∈B such that the interior of Lβ in L is not empty.

Since Mis second countable, B0 is countable.Claim. Sβ∈B0 Lβ is dense in L.Let k1 be the smallest number in δ(L) and let B1 be the set of all β ∈B such that δ(Lβ) = k1.Then by lemma 2.3 each Lβ for β ∈B1 is open.

Let L1 be the closure of Sβ∈B1 Lβ. Let k2be the smallest number in δ(L \ L1) and let B2 be the set of all β ∈B with β(Lβ) = k2 andLβ ∩(L \ L1) ̸= ∅.

Then by lemma 2.3 again Lβ ∩(L \ L1) ̸= ∅is open in L so Lβ has nonempty interior for each β ∈B2. Then let L2 denote the closure of Sβ∈B1∪B2 Lβ and continuethe process.

Since by lemma 2.3 we always find new Lβ with non empty interior, we finallyexhaust L and the claim follows.Let (M 1λ)λ∈C1 be a suitably chosen cover of M by subsets of the sets i−11 (Lβ) such that eachi2| int M 1λ is an embedding for each λ. Let C10 be the set of all λ such that M 1λ has non emptyinterior.

Let similarly (M 2µ)µ∈C2 be a cover for i2. Then there are at most countably many setsM 1λ with λ ∈C10, the union Sλ∈C10 int M 1λ is dense and consequently Sλ∈C10 M 1λ = M; similarlyfor the M 2µ.2.7.

Procedure. Given immersions i1 and i2 as in 2.6 we will try to construct a diffeomor-phism f : M →M with i2 ◦f = i1.

If we meet an obstacle to the construction this will give usenough control on the situation to separate i1 and i2.Choose λ0 ∈C10 so that int M 1λ0 ̸= ∅. Then i1 : int M 1λ0 →Lβ1(λ0) is an embedding, whereβ1 : C1 →B is the mapping satisfying i1(M 1λ) ⊆Lβ1(λ) for all λ ∈C1.

THE SPACE OF IMMERSIONS7Now we choose µ0 ∈β−12 β1(λ0) ⊂C20 such that f := (i2| int M 2µ0)−1 ◦i1| int M 1λ0 is adiffeomorphism int M 1λ0 →int M 2µ0. Note that f is uniquely determined by the choice of µ0, ifit exists, by lemma 1.3.

So we will repeat the following construction for every µ0 ∈β−12 β1(λ0) ⊂C20.Now we try to extend f. We choose λ1 ∈C10 such that M1λ0 ∩M1λ1 ̸= ∅.Case a. Only λ1 = λ0 is possible, so M 1λ0 is dense in M since M is connected and we mayextend f by continuity to a diffeomorphism f : M →M with i2 ◦f = i1.Case b.

We can find λ1 ̸= λ0. We choose x ∈M1λ0 ∩M1λ1 and a sequence (xn) in M 1λ0 withxn →x.

Then we have a sequence (f(xn)) in B.Case ba. y := lim f(xn) exists in M. Then there is µ1 ∈C20 such that y ∈M2µ0 ∩M2µ1.Let U 1α1 be an open neighborhood of x in M such that i1|U 1α1 is an embedding and letsimilarly U 2α2 be an open neighborhood of y in M such that i2|U 2α2 is an embedding.Weconsider now the set i−12 i1(U 1α1).

There are two cases possible.Case baa. The set i−12 i1(U 1α1) is a neighborhood of y.

Then we extend f to i−11 (i1(U 1α1) ∩i2(U 2α2)) by i−12◦i1. Then f is defined on some open subset of int M 1λ1 and by the situationchosen in 2.6 f extends to the whole of int M 1λ1.Case bab.

The set i−12 i1(U 1α1) is not a neighborhood of y. This is a definite obstruction tothe extension of f.Case bb.The sequence (xn) has no limit in M.This is a definite obstruction to theextension of f.If we meet an obstruction we stop and try another µ0.

If for all admissible µ0 we meetobstructions we stop and remember the data. If we do not meet an obstruction we repeat theconstruction with some obvious changes.2.8.

Lemma. The construction of 2.7 in the setting of 2.6 either produces a diffeomorphismf : M →M with i2 ◦f = i1 or we may separate i1 and i2 by open sets in Immprop(M, N) whichare saturated with respect to the action of Diff(M)Proof.

If for some µ0 we do not meet any obstruction in the construction 2.7, the resulting fis defined on the whole of M and it is a continuous mapping M →M with i2 ◦f = i1. Since i1and i2 are locally embeddings, f is smooth and of maximal rank.

Since i1 and i2 are proper, fis proper. So the image of f is open and closed and since M is connected, f is a surjective localdiffeomorphism, thus a covering mapping M →M.

But since δ1 = δ2 the mapping f must bea 1-fold covering, so a diffeomorphism.If for all µ0 ∈β−12 β1(λ0) ⊂C20 we meet obstructions we choose small mutually distinct openneighborhoods V 1λ of the sets i1(M 1λ). We consider the Whitney C0-open neighborhood V1 ofi1 consisting of all immersions j1 with j1(M 1λ) ⊂V 1λ for all λ.

Let V2 be a similar neighborhoodof i2.We claim that V1 ◦Diff(M) and V2 ◦Diff(M) are disjoint.For that it suffices to showthat for any j1 ∈V1 and j2 ∈V2 there does not exist a diffeomorphism f ∈Diff(M) withj2 ◦f = j1. For that to be possible the immersions j1 and j2 must have the same image L andthe same functions δ(j1), δ(j2) : L →N.

But now the combinatorial relations of the slightlydistinct new sets M 1λ, Lβ, and M 2µ are contained in the old ones, so any try to construct sucha diffeomorphism f starting from the same λ0 meets the same obstructions.□

8CERVERA, MASCAR ´O, MICHOR3. Singular orbits3.1.

Let i ∈Imm(M, N) be an immersion which is not free. Then we have a nontrivial isotropysubgroup Diffi(M) ⊂Diff(M) consisting of all f ∈Diff(M) with i ◦f = i.Lemma.

Then the isotropy subgroup Diffi(M) acts properly discontinuously on M, so theprojection q1 : M →M1 := M/ Diffi(M) is a covering map and a submersion for a uniquestructure of a smooth manifold on M1. There is an immersion i1 : M1 →N with i = i1 ◦q1.In particular Diffi(M) is countable, and finite if M is compact.Proof.

We have to show that for each x ∈M there is an open neighborhood U such thatf(U) ∩U = ∅for f ∈Diffi(M) \ {Id}. We consider the setup 1.1 for i.

By the proof of 1.3 wehave f(U iα) ∩U iα = {x ∈U iα : f(x) = x} for any f ∈Diffi(M). If f has a fixed point then by1.3 f = Id, so f(U iα) ∩U iα = ∅for all f ∈Diffi(M) \ {Id}.

The rest is clear.□The factorized immersion i1 is in general not a free immersion. The following is an examplefor that: LetM0 −→α M1 −→β M2 −→γ M3be a sequence of covering maps with fundamental groups 1 →G1 →G2 →G3.

Then thegroup of deck transformations of γ is given by NG3(G2)/G2, the normalizer of G2 in G3,and the group of deck transformations of γ ◦β is NG3(G1)/G1. We can easily arrange thatNG3(G2) ⊈NG3(G1), then γ admits deck transformations which do not lift to M1.

Then wethicken all spaces to manifolds, so that γ ◦β plays the role of the immersion i.3.2.Theorem. Let i ∈Imm(M, N) be an immersion which is not free.Then there is acovering map q2 : M →M2 which is also a submersion such that i factors to an immersioni2 : M2 →N which is free.Proof.

Let q0 : M0 →M be the universal covering of M and consider the immersion i0 =i ◦q0 : M0 →N and its isotropy group Diffi0(M0). By 3.1 it acts properly discontinuously onM0 and we have a submersive covering q02 : M0 →M2 and an immersion i2 : M2 →N withi2 ◦q02 = i0 = i ◦q0.

By comparing the respective groups of deck transformations it is easilyseen that q02 : M0 →M2 factors over q1 ◦q0 : M0 →M →M1 to a covering q12 : M1 →M2.The mapping q2 := q12 ◦q1 : M →M2 is the looked for covering: If f ∈Diff(M2) fixes i2, itlifts to a diffeomorphism f0 ∈Diff(M0) which fixes i0, so is in Diffi0(M0), so f = Id.□3.3. Convention.

In order to avoid complications we assume that from now on M is such amanifold that(1) For any covering M →M1, any diffeomorphism M1 →M1 admits a lift M →M.If M is simply connected, condition (1) is satisfied. Also for M = S1 condition (1) is easilyseen to be valid.

So what follows is applicable to loop spaces.Condition (1) implies that in the proof of 3.2 we have M1 = M2.3.4. Description of a neighborhood of a singular orbit.

Let M be a manifold satisfying3.3.(1). In the situation of 3.1 we consider the normal bundles pi : N(i) →M and pi1 : N(i1) →M1.

Then the covering map q1 : M →M1 lifts uniquely to a vector bundle homomorphismN(q1) : N(i) →N(i1) which is also a covering map, such that τ i1 ◦N(q1) = τ i.

THE SPACE OF IMMERSIONS9We have M1 = M/ Diffi(M) and the group Diffi(M) acts also as the group of deck transfor-mations of the covering N(q1) : N(i) →N(i1) by Diffi(M) ∋f 7→N(f), whereN(i) −−−−→N(f)N(i)yyM−−−−→fMis a vector bundle isomorphism for each f ∈Diffi(M). If we equip N(i) and N(i1) with thefiber Riemann metrics induced from the fixed Riemannian metric g on N, the mappings N(q1)and all N(f) are fiberwise linear isometries.Let us now consider the right action of Diffi(M) on the space of sections C∞c (N(i)) givenby f ∗s := N(f)−1 ◦s ◦f.From the proof of theorem 1.5 we recall now the setsC∞(M, N(i)) ⊃V(i) ←−−−−ϕiU(i)xxC∞c (N(i)) ⊃C∞c (U i)ϕi←−−−−Q(i).All horizontal mappings are again diffeomorphisms and the vertical mappings are inclusions.But since the action of Diff(M) on i is not free we cannot extend the splitting submanifold Q(i)to an orbit cylinder as we did in the proof on theorem 1.5.

Q(i) is again a smooth transversalfor the orbit though i.For any f ∈Diff(M) and s ∈C∞c (U i) ⊂C∞c (N(i)) we haveϕ−1i (f ∗s) = τi∗(f ∗s) = τ i∗(s) ◦f.So the space q∗1C∞c (N(i1)) of all sections of N(i) →M which factor to sections of N(i1) →M1,is exactly the space of all fixed points of the action of Diffi(M) on C∞c (N(i)); and they aremapped by τ i∗= ϕ−1ito immersions in Q(i) which have again Diffi(M) as isotropy group.If s ∈C∞c (U i) ⊂C∞c (N(i)) is an arbitrary section, the orbit through τ i∗(s) ∈Q(i) hits thetransversal Q(i) again in the points τi∗(f ∗s) for f ∈Diffi(M).We summarize all this in the following theorem:3.5. Theorem.

Let M be a manifold satisfying condition (1) of 3.3. Let i ∈Imm(M, N) bean immersion which is not free, i.e.

has non trivial isotropy group Diffi(M).Then in the setting and notation of 3.4 in the following commutative diagram the bottommappingImmfree(M1, N)(q1)∗−−−−→Imm(M, N)πyyπImmfree(M1, N)/ Diff(M1) −−−−→Imm(M, N)/ Diff(M)is the inclusion of a (possibly non Hausdorff) manifold, the stratum of π(i) in the stratificationof the orbit space. This stratum consists of the orbits of all immersions which have Diffi(M)as isotropy group.

10CERVERA, MASCAR ´O, MICHOR3.6. The orbit structure.

We have the following description of the orbit structure near i inImm(M, N): For fixed f ∈Diffi(M) the set of fixed points Fix(f) := {j ∈Q(i) : j ◦f = j} iscalled a generalized wall. The union of all generalized walls is called the diagram D(i) of i. Aconnected component of the complement Q(i) \ D(i) is called a generalized Weyl chamber.

Thegroup Diffi(M) maps walls to walls and chambers to chambers. The immersion i lies in everywall.We shall see shortly that there is only one chamber and that the situation is rather distinctfrom that of reflection groups.If we view the diagram in the space C∞c (U i) ⊂C∞c (N(i)) which is diffeomorphic to Q(i),then it consist of traces of closed linear subspaces, because the action of Diffi(M) on C∞c (N(i))consists of linear isometries in the following way.

Let us tensor the vector bundle N(i) →Mwith the natural line bundle of half densities on M, and let us remember one positive halfdensity to fix an isomorphism with the original bundle. Then Diffi(M) still acts on this newbundle N1/2(i) →M and the pullback action on sections with compact support is isometric forthe inner product⟨s1, s2⟩:=ZMg(s1, s2).We consider the walls and chambers now extended to the whole space in the obvious manner.3.7.

Lemma. Each wall in C∞c (N1/2(i)) is a closed linear subspace of infinite codimension.Since there are at most countably many walls, there is only one chamber.Proof.

From the proof of lemma 3.1 we know that f(U iα) ∩U iα = ∅for all f ∈Diffi(M) and allsets U iα from the setup 1.1. Take a section s in the wall of fixed points of f. Choose a section sαwith support in some U iα and let the section s be defined by s|U iα = sα|U iα, s|f −1(U iα) = −f ∗sα,0 elsewhere.

Then obviously ⟨s, s′⟩= 0 for all s′ in the wall of f. But this construction furnishesan infinite dimensional space contained in the orthogonal complement of the wall of f.□ReferencesBinz, Ernst; Fischer, Hans R., The manifold of embeddings of a closed manifold, Proc. Differential geometricmethods in theoretical physics, Clausthal 1978, Springer Lecture Notes in Physics 139, 1981.Fr¨olicher, A.; Kriegl, A., Linear spaces and differentiation theory, Pure and Applied Mathematics, J. Wiley,Chichester, 1988.Kriegl, A.; Michor, P. W., A convenient setting for real analytic mappings, 52 p., to appear, Acta Mathematica(1990).Michor, P. W., Manifolds of smooth maps, Cahiers Topol.

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Diff. 21 (1980a), 63–86.Michor, P. W., Manifolds of smooth maps III: The principal bundle of embeddings of a non compact smoothmanifold, Cahiers Topol.

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21 (1980b), 325–337.Michor, P. W., Manifolds of differentiable mappings, Shiva, Orpington, 1980c.Michor, P. W., Manifolds of smooth mappings IV: Theorem of De Rham, Cahiers Top. Geo.

Diff. 24 (1983),57–86.Michor, P. W., Gauge theory for diffeomorphism groups, Proceedings of the Conference on Differential GeometricMethods in Theoretical Physics, Como 1987, K. Bleuler and M. Werner (eds.

), Kluwer, Dordrecht, 1988,pp. 345–371.Weinstein, Alan, Symplectic manifolds and their Lagrangian manifolds, Advances in Math.

6 (1971), 329–345.Departamento de Geometr´ıa y Topolog´ıa, Facultad de Matem´aticas, Universidad de Valencia,E-46100 BURJASSOT, VALENCIA, SPAIN

THE SPACE OF IMMERSIONS11E-mail address: mascaro@evalun11.bitnetInstitut f¨ur Mathematik, Universit¨at Wien, Strudlhofgasse 4, A-1090 Wien, Austria

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