---

Differential Geometry and its Applications

arXiv:math/9202206v1 [math.DG] 1 Feb 1992Differential Geometry and its Applications1 (1991), 159–176ASPECTS OF THE THEORY OFINFINITE DIMENSIONAL MANIFOLDSAndreas KrieglPeter W. MichorInstitut f¨ur Mathematik, Universit¨at Wien,Strudlhofgasse 4, A-1090 Wien, Austria.September 2, 1989Abstract. The convenient setting for smooth mappings, holomorphic mappings, andreal analytic mappings in infinite dimension is sketched.

Infinite dimensional manifoldsare discussed with special emphasis on smooth partitions of unity and tangent vectors asderivations. Manifolds of mappings and diffeomorphisms are treated.

Finally the differ-ential structure on the inductive limits of the groups GL(n), SO(n) and some of theirhomogeneus spaces is treated.IntroductionThe theory of infinite dimensional manifolds has already a long history. In the fiftiesand sixties smooth manifolds modeled on Banach spaces were investigated a lot.

Herethe starting point were the investigations of Marston Morse on the index of geodesics inRiemannian manifolds. He used Hilbert manifolds of curves in a Riemannian manifold.Later on Fr´echet manifolds were investigated from the point of view of topology: itwas shown that under certain weak conditions they could be embedded as open subsetsin the model space.Then starting with a seminal short paper of J. Eells began the investigation of man-ifolds of mappings.But all this became important for the mainstream of mathematics when loop groupsand their Lie algebras - the Kac-Moody Lie algebras were used in Physics.In this review paper we will try to find our own way through the field of infinitedimensional manifolds, and we will concentrate on the smooth manifolds, and on thosewhich are not modeled on Banach spaces or Hilbert spaces - although the latter are veryimportant as a technical mean to prove very important theorems like those leading toexotic R4’s.1991 Mathematics Subject Classification.

26E15, 26E05, 46G20, 58B10, 58B12, 58D05, 58D10,58D15.Key words and phrases. Convenient vector spaces, manifolds of mappings, diffeomorphism groups,infinite dimensional Lie groups and Grassmann manifolds.This paper is in final form and no version of it will appear elsewhereTypeset by AMS-TEX1

2ANDREAS KRIEGL PETER W. MICHORThe material presented in the later sections is from [Kriegl-Michor, Foundations ofGlobal Analysis].Table of contents1. Calculus of smooth mappings2.

Calculus of holomorphic mappings3. Calculus of real analytic mappings4.

Infinite dimensional manifolds5. Manifolds of mappings6.

Manifolds for algebraic topology1. Calculus of smooth mappings1.1.

The traditional differential calculus works well for finite dimensional vector spacesand for Banach spaces. For more general locally convex spaces a whole flock of differenttheories were developed, each of them rather complicated and none really convincing.The main difficulty is that the composition of linear mappings stops to be jointly contin-uous at the level of Banach spaces, for any compatible topology.

This was the originalmotivation for the development of a whole new field within general topology, conver-gence spaces.Then in 1982, Alfred Fr¨olicher and Andreas Kriegl presented independently the solu-tion to the question for the right differential calculus in infinite dimensions. They joinedforces in the further development of the theory and the (up to now) final outcome isthe book [Fr¨olicher-Kriegl, 1988].In this section we will sketch the basic definitions and the most important results ofthe Fr¨olicher-Kriegl calculus.1.2.

The c∞-topology. Let E be a locally convex vector space.

A curve c : R →Eis called smooth or C∞if all derivatives exist and are continuous - this is a conceptwithout problems. Let C∞(R, E) be the space of smooth functions.

It can be shown thatC∞(R, E) does not depend on the locally convex topology of E, only on its associatedbornology (system of bounded sets).The final topologies with respect to the following sets of mappings into E coincide:(1) C∞(R, E). (2) Lipschitz curves (so that { c(t)−c(s)t−s: t ̸= s} is bounded in E).

(3) {EB →E : B bounded absolutely convex in E}, where EB is the linear span ofB equipped with the Minkowski functional pB(x) := inf{λ > 0 : x ∈λB}. (4) Mackey-convergent sequences xn →x (there exists a sequence 0 < λn ր ∞with λn(xn −x) bounded).This topology is called the c∞-topology on E and we write c∞E for the resultingtopological space.

In general (on the space D of test functions for example) it is finerthan the given locally convex topology, it is not a vector space topology, since scalarmultiplication is no longer jointly continuous.The finest among all locally convextopologies on E which are coarser than c∞E is the bornologification of the given locallyconvex topology. If E is a Fr´echet space, then c∞E = E.

ASPECTS OF THE THEORY OF INFINITE DIMENSIONAL MANIFOLDS31.3. Convenient vector spaces.

Let E be a locally convex vector space. E is saidto be a convenient vector space if one of the following equivalent conditions is satisfied(called c∞-completeness):(1) Any Mackey-Cauchy-sequence (so that (xn −xm) is Mackey convergent to 0)converges.

(2) If B is bounded closed absolutely convex, then EB is a Banach space. (3) Any Lipschitz curve in E is locally Riemann integrable.

(4) For any c1 ∈C∞(R, E) there is c2 ∈C∞(R, E) with c1 = c′2 (existence ofantiderivative).1.4.Lemma. Let E be a locally convex space.Then the following properties areequivalent:(1) E is c∞-complete.

(2) If f : R →E is scalarwise Lipk, then f is Lipk, for k > 1. (3) If f : R →E is scalarwise C∞then f is differentiable at 0.

(4) If f : R →E is scalarwise C∞then f is C∞.Here a mapping f : R →E is called Lipk if all partial derivatives up to order kexist and are Lipschitz, locally on R. f scalarwise C∞means that λ ◦f is C∞for allcontinuous linear functionals on E.This lemma says that on a convenient vector space one can recognize smooth curvesby investigating compositions with continuous linear functionals.1.5. Smooth mappings.

Let E and F be locally convex vector spaces. A mappingf : E →F is called smooth or C∞, if f ◦c ∈C∞(R, F) for all c ∈C∞(R, E); sof∗: C∞(R, E) →C∞(R, F) makes sense.

Let C∞(E, F) denote the space of all smoothmapping from E to F.For E and F finite dimensional this gives the usual notion of smooth mappings: thishas been first proved in [Boman, 1967]. Constant mappings are smooth.

Multilinearmappings are smooth if and only if they are bounded. Therefore we denote by L(E, F)the space of all bounded linear mappings from E to F.1.6.

Structure on C∞(E, F). We equip the space C∞(R, E) with the bornologifi-cation of the topology of uniform convergence on compact sets, in all derivatives sep-arately.

Then we equip the space C∞(E, F) with the bornologification of the initialtopology with respect to all mappings c∗: C∞(E, F) →C∞(R, F), c∗(f) := f ◦c, forall c ∈C∞(R, E).1.7. Lemma.

For locally convex spaces E and F we have:(1) If F is convenient, then also C∞(E, F) is convenient, for any E. The spaceL(E, F) is a closed linear subspace of C∞(E, F), so it also is convenient. (2) If E is convenient, then a curve c : R →L(E, F) is smooth if and only ift 7→c(t)(x) is a smooth curve in F for all x ∈E.1.8.Theorem.

The category of convenient vector spaces and smooth mappings iscartesian closed. So we have a natural bijectionC∞(E × F, G) ∼= C∞(E, C∞(F, G)),which is even a diffeomorphism.Of course this statement is also true for c∞-open subsets of convenient vector spaces.

4ANDREAS KRIEGL PETER W. MICHOR1.9. Corollary.

Let all spaces be convenient vector spaces. Then the following canon-ical mappings are smooth.ev : C∞(E, F) × E →F,ev(f, x) = f(x)ins : E →C∞(F, E × F),ins(x)(y) = (x, y)()∧: C∞(E, C∞(F, G)) →C∞(E × F, G)()∨: C∞(E × F, G) →C∞(E, C∞(F, G))comp : C∞(F, G) × C∞(E, F) →C∞(E, G)C∞(,) : C∞(F, F ′) × C∞(E′, E) →C∞(C∞(E, F), C∞(E′, F ′))(f, g) 7→(h 7→f ◦h ◦g)Y:YC∞(Ei, Fi) →C∞(YEi,YFi)1.10.Theorem.

Let E and F be convenient vector spaces.Then the differentialoperatord : C∞(E, F) →C∞(E, L(E, F)),df(x)v := limt→0f(x + tv) −f(x)t,exists and is linear and bounded (smooth). Also the chain rule holds:d(f ◦g)(x)v = df(g(x))dg(x)v.1.11.Remarks.

Note that the conclusion of theorem 1.8 is the starting point ofthe classical calculus of variations, where a smooth curve in a space of functions wasassumed to be just a smooth function in one variable more.If one wants theorem 1.8 to be true and assumes some other obvious properties, thenthe calculus of smooth functions is already uniquely determined.There are, however, smooth mappings which are not continuous. This is unavoidableand not so horrible as it might appear at first sight.

For example the evaluation E×E′ →R is jointly continuous if and only if E is normable, but it is always smooth. Clearlysmooth mappings are continuous for the c∞-topology.For Fr´echet spaces smoothness in the sense described here coincides with the notionC∞cof [Keller, 1974].

This is the differential calculus used by [Michor, 1980], [Milnor,1984], and [Pressley-Segal, 1986].A prevalent opinion in contemporary mathematics is, that for infinite dimensionalcalculus each serious application needs its own foundation. By a serious application oneobviously means some application of a hard inverse function theorem.

These theoremscan be proved, if by assuming enough a priori estimates one creates enough Banachspace situation for some modified iteration procedure to converge. Many authors tryto build their platonic idea of an a priori estimate into their differential calculus.

Wethink that this makes the calculus inapplicable and hides the origin of the a prioriestimates. We believe, that the calculus itself should be as easy to use as possible, andthat all further assumptions (which most often come from ellipticity of some nonlinear

ASPECTS OF THE THEORY OF INFINITE DIMENSIONAL MANIFOLDS5partial differential equation of geometric origin) should be treated separately, in a settingdepending on the specific problem. We are sure that in this sense the Fr¨olicher-Krieglcalculus as presented here and its holomorphic and real analytic offsprings in sections2 and 3 below are universally usable for most applications.We believe that the recent development of the theory of locally convex spaces missedits original aim: the development of calculus.

It laid too much emphasis on (locallyconvex) topologies and ignored and denigraded the work of Hogbe-Nlend, Colombeauand collaborators on the original idea of Sebasti˜ao e Silva, that bornologies are the rightconcept for this kind of functional analysis.2. Calculus of holomorphic mappings2.1.

Along the lines of thought of the Fr¨olicher-Kriegl calculus of smooth mappings, in[Kriegl-Nel, 1985] the cartesian closed setting for holomorphic mappings was developed.The right definition of this calculus was already given by [Fantappi´e, 1930 and 1933]. Wewill now sketch the basics and the main results.

It can be shown that again convenientvector spaces are the right ones to consider. Here we will start with them for the sakeof shortness.2.2.

Let E be a complex locally convex vector space whose underlying real space isconvenient – this will be called convenient in the sequel. Let D ⊂C be the open unitdisk and let us denote by H(D, E) the space of all mappings c : D →E such thatλ ◦c : D →C is holomorphic for each continuous complex-linear functional λ on E. Itselements will be called the holomorphic curves.If E and F are convenient complex vector spaces (or c∞-open sets therein), a mappingf : E →F is called holomorphic if f ◦c is a holomorphic curve in F for each holomorphiccurve c in E. Obviously f is holomorphic if and only if λ ◦f : E →C is holomorphicfor each complex linear continuous functional λ on F. Let H(E, F) denote the space ofall holomorphic mappings from E to F.2.3.Theorem (Hartog’s theorem).

Let Ek for k = 1, 2 and F be complex con-venient vector spaces and let Uk ⊂Ek be c∞-open. A mapping f : U1 × U2 →F isholomorphic if and only if it is separately holomorphic (i. e. f(, y) and f(x,) areholomorphic for all x ∈U1 and y ∈U2).This implies also that in finite dimensions we have recovered the usual definition.2.4 Lemma.

If f : E ⊃U →F is holomorphic then df : U × E →F exists, isholomorphic and C-linear in the second variable.A multilinear mapping is holomorphic if and only if it is bounded.2.5 Lemma. If E and F are Banach spaces and U is open in E, then for a mappingf : U →F the following conditions are equivalent:(1) f is holomorphic.

(2) f is locally a convergent series of homogeneous continuous polynomials. (3) f is C-differentiable in the sense of Fr´echet.2.6 Lemma.

Let E and F be convenient vector spaces. A mapping f : E →F isholomorphic if and only if it is smooth and its derivative is everywhere C-linear.

6ANDREAS KRIEGL PETER W. MICHORAn immediate consequence of this result is that H(E, F) is a closed linear subspaceof C∞(ER, FR) and so it is a convenient vector space if F is one, by 1.7. The chain rulefollows from 1.10.

The following theorem is an easy consequence of 1.8.2.7 Theorem. The category of convenient complex vector spaces and holomorphic map-pings between them is cartesian closed, i. e.H(E × F, G) ∼= H(E, H(F, G)).An immediate consequence of this is again that all canonical structural mappings asin 1.9 are holomorphic.3.

Calculus of real analytic mappings3.1. In this section we sketch the cartesian closed setting to real analytic mappings ininfinite dimension following the lines of the Fr¨olicher-Kriegl calculus, as it is presentedin [Kriegl-Michor, 1990].

Surprisingly enough one has to deviate from the most obviousnotion of real analytic curves in order to get a meaningful theory, but again convenientvector spaces turn out to be the right kind of spaces.3.2. Real analytic curves.

Let E be a real convenient vector space with dual E′. Acurve c : R →E is called real analytic if λ ◦c : R →R is real analytic for each λ ∈E′.It turns out that the set of these curves depends only on the bornology of E.In contrast a curve is called topologically real analytic if it is locally given by powerseries which converge in the topology of E. They can be extended to germs of holomor-phic curves along R in the complexification EC of E. If the dual E′ of E admits a Bairetopology which is compatible with the duality, then each real analytic curve in E is infact topologically real analytic for the bornological topology on E.3.3.

Real analytic mappings. Let E and F be convenient vector spaces.

Let U bea c∞-open set in E. A mapping f : U →F is called real analytic if and only if it issmooth (maps smooth curves to smooth curves) and maps real analytic curves to realanalytic curves.Let Cω(U, F) denote the space of all real analytic mappings. We equip the spaceCω(U, R) of all real analytic functions with the initial topology with respect to thefamilies of mappingsCω(U, R)c∗−→Cω(R, R), for all c ∈Cω(R, U)Cω(U, R)c∗−→C∞(R, R), for all c ∈C∞(R, U),where C∞(R, R) carries the topology of compact convergence in each derivative sep-arately as in section 1, and where Cω(R, R) is equipped with the final locally convextopology with respect to the embeddings (restriction mappings) of all spaces of holo-morphic mappings from a neighborhood V of R in C mapping R to R, and each of thesespaces carries the topology of compact convergence.Furthermore we equip the space Cω(U, F) with the initial topology with respect tothe family of mappingsCω(U, F)λ∗−→Cω(U, R), for all λ ∈F ′.It turns out that this is again a convenient space.

ASPECTS OF THE THEORY OF INFINITE DIMENSIONAL MANIFOLDS73.4. Theorem.

In the setting of 3.3 a mapping f : U →F is real analytic if and onlyif it is smooth and is real analytic along each affine line in E.3.5. Theorem.

The category of convenient spaces and real analytic mappings is carte-sian closed. So the equationCω(U, Cω(V, F)) ∼= Cω(U × V, F)is valid for all c∞-open sets U in E and V in F, where E, F, and G are convenientvector spaces.This implies again that all structure mappings as in 1.9 are real analytic.

Furthermorethe differential operatord : Cω(U, F) →Cω(U, L(E, F))exists, is unique and real analytic. Multilinear mappings are real analytic if and only ifthey are bounded.

Powerful real analytic uniform boundedness principles are available.4. Infinite dimensional manifolds4.1.

In this section we will concentrate on two topics: Smooth partitions of unity, andseveral kinds of tangent vectors.4.2. In the usual way we define manifolds by gluing c∞-open sets in convenient vectorspaces via smooth (holomorphic, real analytic) diffeomorphisms.

Then we equip themwith the identification topology with respect to the c∞-topologies on all modeling spaces.We require some properties from this topology, like Hausdorffand regular (which hereis not a consequence of Hausdorff).Mappings between manifolds are smooth (holomorphic, real analytic), if they havethis property when composed which any chart mappings.4.3. Lemma.

A manifold M is metrizable if and only if it is paracompact and modeledon Fr´echet spaces.4.4. Lemma.

For a convenient vector space E the set C∞(M, E) of smooth E-valuedfunctions on a manifold M is again a convenient vector space. Likewise for the realanalytic and holomorphic case.4.5.

Theorem. If M is a smooth manifold modeled on convenient vector spaces ad-mitting smooth bump functions and U is a locally finite open cover of M, then thereexists a smooth partition of unity {ϕU : U ∈U} with carr/supp(ϕU) ⊂U for all U ∈U.If M is in addition paracompact, then this is true for every open cover U of M.Convenient vector spaces which are nuclear admit smooth bump functions.4.6.

The tangent spaces of a convenient vector space E. Let a ∈E. A kinematictangent vector with foot point a is simply a pair (a, X) with X ∈E.

Let TaE = E bethe space of all kinematic tangent vectors with foot point a. It consists of all derivativesc′(0) at 0 of smooth curves c : R →E with c(0) = a, which explains the choice of thename kinematic.For each open neighborhood U of a in E (a, X) induces a linear mapping Xa :C∞(U, R) →R by Xa(f) := df(a)(Xa), which is continuous for the convenient vector

8ANDREAS KRIEGL PETER W. MICHORspace topology on C∞(U, R), and satisfies Xa(f · g) = Xa(f) · g(a) + f(a) · Xa(g), so itis a continuous derivation over eva. The value Xa(f) depends only on the germ of f ata.An operational tangent vector of E with foot point a is a bounded derivation ∂:C∞a (E, R) →R over eva.

Let DaE be the vector space of all these derivations. Any ∂∈DaE induces a bounded derivation C∞(U, R) →R over eva for each open neighborhoodU of a in E. So the vector space DaE is a closed linear subspace of the convenientvector space QU L(C∞(U, R), R).

We equip DaE with the induced convenient vectorspace structure. Note that the spaces DaE are isomorphic for all a ∈E.Example.

Let Y ∈E′′ be an element in the bidual of E. Then for each a ∈E we havean operational tangent vector Ya ∈DaE, given by Ya(f) := Y (df(a)). So we have acanonical injection E′′ →DaE.Example.

Let ℓ: L2(E; R) →R be a bounded linear functional which vanishes on thesubset E′ ⊗E′. Then for each a ∈E we have an operational tangent vector ∂2ℓ|a ∈DaEgiven by ∂2ℓ|a(f) := ℓ(d2f(a)), sinceℓ(d2(fg)(a)) == ℓ(d2f(a)g(a) + df(a) ⊗dg(a) + dg(a) ⊗df(a) + f(a)d2g(a))= ℓ(d2f(a))g(a) + 0 + f(a)ℓ(d2g(a)).4.7.

Lemma. Let ℓ∈Lksym(E; R)′ be a bounded linear functional which vanishes onthe subspacek−1Xi=1Lisym(E; R) ∨Lk−isym(E; R)of decomposable elements of Lksym(E; R).

Then ℓdefines an operational tangent vector∂kℓ|a ∈DaE for each a ∈E by∂kℓ|a(f) := ℓ(dkf(a)).The linear mapping ℓ7→∂kℓ|a is an embedding onto a topological direct summand D(k)a Eof DaE.Its left inverse is given by ∂7→(Φ 7→∂((Φ ◦diag)(a+))).The sumPk>0 D(k)a E in DaE is a direct one.4.8.Lemma. If E is an infinite dimensional Hilbert space, all operational tangentspace summands D(k)0 E are not zero.4.9.

Definition. A convenient vector space is said to have the (bornological) approxi-mation property if E′⊗E is dense in L(E, E) in the bornological locally convex topology.The following spaces have the bornological approximation property: R(N), nuclearFr´echet spaces, nuclear (LF) spaces.4.10 Theorem.

Let E be a convenient vector space which has the approximation prop-erty. Then we have DaE = D(1)a E ∼= E′′.

So if E is in addition reflexive, each opera-tional tangent vector is a kinematic one.

ASPECTS OF THE THEORY OF INFINITE DIMENSIONAL MANIFOLDS94.11. The kinematic tangent bundle TM of a manifold M is constructed by gluing allthe kinematic tangent bundles of charts with the help of the kinematic tangent mappings(derivatives) of the chart changes.

TM →M is a vector bundle and T : C∞(M, N) →C∞(TM, TN) is well defined and has the usual properties.4.12. The operational tangent bundle DM of a manifold M is constructed by gluingall operational tangent spaces of charts.

Then πM : DM →M is again a vector bundlewhich contains the kinematic tangent bundle TM as a splitting subbundle. Also for eachk ∈N the same gluing construction as above gives us tangent bundles D(k)M which aresplitting sub bundles of DM.

The mappings D(k) : C∞(M, N) →C∞(D(k)M, D(k)N)are well defined for all k (including no k) and have the usual properties.Note that for manifolds modeled on reflexive spaces having the bornological approx-imation property the operational and the kinematic tangent bundles coincide.5. Manifolds of mappings5.1.

Theorem (Manifold structure of C∞(M, N)). Let M and N be smooth finitedimensional manifolds, let M be compact.

Then the space C∞(M, N) of all smoothmappings from M to N is a smooth manifold, modeled on spaces C∞(f ∗TN) of smoothsections of pullback bundles along f : M →N over M.A careful description of this theorem (but without the help of the Fr¨olicher-Krieglcalculus) can be found in [Michor, 1980]. We include a proof of this result here becausethe result is important and the proof is much simpler now.Proof.

Choose a smooth Riemannian metric on N. Let exp : TN ⊇U →N be thesmooth exponential mapping of this Riemannian metric, defined on a suitable openneighborhood of the zero section. We may assume that U is chosen in such a way that(πN, exp) : U →N × N is a smooth diffeomorphism onto an open neighborhood V ofthe diagonal.For f ∈C∞(M, N) we consider the pullback vector bundleM ×N TNf ∗TNπ∗Nf−−−−→TNf ∗πNyyπNMf−−−−→N.Then C∞(f ∗TN) is canonically isomorphic to C∞f (M, TN) := {h ∈C∞(M, TN) :πN ◦h = f} via s 7→(π∗Nf) ◦s and (IdM, h) ←h.

We consider the space C∞c (f ∗TN) ofall smooth sections with compact support and equip it with the inductive limit topologyC∞c (f ∗TN) = inj limKC∞K (f ∗TN),where K runs through all compact sets in M and each of the spaces C∞K (f ∗TN) isequipped with the topology of uniform convergence (on K) in all derivatives separately.Now letUf := {g ∈C∞(M, N) : (f(x), g(x)) ∈V for all x ∈M, g ∼f},uf : Uf →C∞c (f ∗TN),uf(g)(x) = (x, exp−1f(x)(g(x))) = (x, ((πN, exp)−1 ◦(f, g))(x)).

10ANDREAS KRIEGL PETER W. MICHORHere g ∼f means that g equals f offsome compact set. Then uf is a bijective mappingfrom Uf onto the set {s ∈C∞c (f ∗TN) : s(M) ⊆f ∗U}, whose inverse is given byu−1f (s) = exp ◦(π∗Nf)◦s, where we view U →N as fiber bundle.

The set uf(Uf) is openin C∞c (f ∗TN) for the topology described above.Now we consider the atlas (Uf, uf)f∈C∞(M,N) for C∞(M, N). Its chart change map-pings are given for s ∈ug(Uf ∩Ug) ⊆C∞c (g∗TN) by(uf ◦u−1g )(s) = (IdM, (πN, exp)−1 ◦(f, exp ◦(π∗Ng) ◦s))= (τ −1f◦τg)∗(s),where τg(x, Yg(x)) := (x, expg(x)(Yg(x)))) is a smooth diffeomorphism τg : g∗TN ⊇g∗U →(g × IdN)−1(V ) ⊆M × N which is fiber respecting over M.Smooth curves in C∞c (f ∗TN) are just smooth sections of the bundle pr∗2f ∗TN →R × M, which have compact support in M locally inR.

The chart change uf ◦u−1g=(τ −1f◦τg)∗is defined on an open subset and obviously maps smooth curves to smoothcurves, therefore it is also smooth.Finally we put the identification topology from this atlas onto the space C∞(M, N),which is obviously finer than the compact open topology and thus Hausdorff.The equation uf ◦u−1g= (τ −1f◦τg)∗shows that the smooth structure does not dependon the choice of the smooth Riemannian metric on N.□5.2. Theorem (Cω-manifold structure of Cω(M, N)).

Let M and N be real ana-lytic manifolds, let M be compact. Then the space Cω(M, N) of all real analytic map-pings from M to N is a real analytic manifold, modeled on spaces Cω(f ∗TN) of realanalytic sections of pullback bundles along f : M →N over M.The proof can be found in [Kriegl-Michor, 1990].

It is a variant of the above proof,using a real analytic Riemannian metric.5.3.Theorem (Cω-manifold structure on C∞(M, N)). Let M and N be realanalytic manifolds, with M compact.

Then the smooth manifold C∞(M, N) is even areal analytic manifold.Proof. For a fixed real analytic exponential mapping on N the charts (Uf, uf) from5.1 for f ∈Cω(M, N) form a smooth atlas for C∞(M, N), since Cω(M, N) is dense inC∞(M, N) by [Grauert, 1958, Proposition 8].

The chart changings uf ◦u−1g= (τ −1f◦τg)∗are real analytic: this follows from a careful description of the set of real analytic curvesinto C∞(f ∗TN). See again [Kriegl-Michor, 1990, 7.7] for more details.□5.4 Remark.

If M is not compact, Cω(M, N) is dense in C∞(M, N) for the Whitney-C∞-topology by [Grauert, 1958, Prop. 8].

This is not the case for the topology used in5.1 in which C∞(M, N) is a smooth manifold. The charts Uf for f ∈Cω(M, N) do notcover C∞(M, N).5.5.

Theorem. Let M and N be smooth manifolds.

Then the two infinite dimensionalsmooth vector bundles TC∞(M, N) and C∞(M, TN) over C∞(M, N) are canonicallyisomorphic. The same assertion is true for Cω(M, N), if M is compact.

ASPECTS OF THE THEORY OF INFINITE DIMENSIONAL MANIFOLDS115.6.Theorem (Exponential law). Let M be a (possibly infinite dimensional)smooth manifold, and let M and N be finite dimensional smooth manifolds.Then we have a canonical embeddingC∞(M, C∞(M, N)) ⊆C∞(M × M, N),where we have equality if and only if M is compact.If M and N are real analytic manifolds with M compact we haveCω(M, Cω(M, N)) = Cω(M × M, N)for each real analytic (possibly infinite dimensional) manifold.5.7.

Corollary. If M is compact and M, N are finite dimensional smooth manifolds,then the evaluation mapping ev : C∞(M, N) × M →N is smooth.If P is another compact smooth manifold, then the composition mapping comp :C∞(M, N) × C∞(P, M) →C∞(P, N) is smooth.In particular f∗: C∞(M, N) →C∞(M, N ′) and g∗: C∞(M, N) →C∞(P, N) aresmooth for f ∈C∞(N, N ′) and g ∈C∞(P, M).The corresponding statement for real analytic mappings is also true.5.8.Theorem (Diffeomorphism groups).

For a smooth manifold M the groupDiff(M) of all smooth diffeomorphisms of M is an open submanifold of C∞(M, M),composition and inversion are smooth.The Lie algebra of the smooth infinite dimensional Lie group Diff(M) is the con-venient vector space C∞c (TM) of all smooth vector fields on M with compact sup-port, equipped with the negative of the usual Lie bracket.The exponential mappingExp : C∞c (TM) →Diff∞(M) is the flow mapping to time 1, and it is smooth.For a compact real analytic manifold M the group Diffω(M) of all real analytic diffeo-morphisms is a real analytic Lie group with Lie algebra Cω(TM) and with real analyticexponential mapping.5.9.Remarks. The group Diff(M) of smooth diffeomorphisms does not carry anyreal analytic Lie group structure by [Milnor, 1984, 9.2], and it has no complexificationin general, see [Pressley-Segal, 1986, 3.3].

The mappingAd ◦Exp : C∞c (TM) →Diff(M) →L(C∞(TM), C∞(TM))is not real analytic, see [Michor, 1983, 4.11].For x ∈M the mapping evx ◦Exp : C∞c (TM) →Diff(M) →M is not real analyticsince (evx ◦Exp)(tX) = FlXt (x), which is not real analytic in t for general smooth X.The exponential mapping Exp : C∞c (TM) →Diff(M) is in a very strong sense notsurjective: In [Grabowski, 1988] it is shown, that Diff(M) contains an arcwise connectedfree subgroup on 2ℵ0 generators which meets the image of Exp only at the identity.The real analytic Lie group Diffω(M) is regular in the sense of [Milnor, 1984. 7.6],who weakened the original concept of [Omori, 1982].

This condition means that themapping associating the evolution operator to each time dependent vector field on Mis smooth. It is even real analytic, compare the proof of theorem 5.9.

12ANDREAS KRIEGL PETER W. MICHOR5.10. Theorem.

Let M and N be smooth manifolds. Then the diffeomorphism groupDiff(M) acts smoothly from the right on the smooth manifold Imm(M, N) of all smoothimmersions M →N, which is an open subset of C∞(M, N).Then the space of orbits Imm(M, N)/ Diff(M) is Hausdorffin the quotient topology.Let Immfree(M, N) be set of all immersions, on which Diff(M) acts freely.

Then thisis open in C∞(M, N) and is the total space of a smooth principal fiber bundleImmfree(M, N) →Immfree(M, N)/ Diff(M).In particular the space Emb(M, N) of all smooth embeddings is the total space ofsmooth principal fiber bundle.This is proved in [Cervera-Mascaro-Michor, 1989], where also the existence of smoothtransversals to each orbit is shown and the stratification of the orbit space into smoothmanifolds is given.5.11. Theorem (Principal bundle of embeddings).

Let M and N be real analyticmanifolds with M compact. Then the set Embω(M, N) of all real analytic embeddingsM →N is an open submanifold of Cω(M, N).

It is the total space of a real analyticprincipal fiber bundle with structure group Diffω(M), whose real analytic base manifoldis the space of all submanifolds of N of type M.See [Kriegl-Michor, 1990], section 6.5.12.Theorem (Classifying space for Diff(M)). Let M be a compact smoothmanifold.

Then the space Emb(M, ℓ2) of smooth embeddings of M into the Hilbert spaceℓ2 is the total space of a smooth principal fiber bundle with structure group Diff(M) andsmooth base manifold B(M, ℓ2), which is a classifying space for the Lie group Diff(M).It carries a universal Diff(M)-connection.In other words:(Emb(M, ℓ2) ×Diff(M) M →B(M, ℓ2)classifies fiber bundles with typical fiber M and carries a universal (generalized) connec-tion.See [Michor, 1988, section 6].6. Manifolds for algebraic topology6.1 Convention.

In this section the space R(N) of all finite sequences with the directsum topology will be denoted by R∞following the common usage in algebraic topology.It is a convenient vector space.We consider on it the weak inner product ⟨x, y⟩:= P xiyi, which is bilinear andbounded, therefore smooth. It is called weak, since it is non degenerate in the followingsense: the associated linear mapping R∞→(R∞)′ = RN is injective, but far from beingsurjective.

We will also use the weak Euclidean distance |x| :=p⟨x, x⟩, whose squareis a smooth function.6.2. Example: The sphere S∞.The sphere S∞is the set {x ∈R∞: ⟨x, x⟩= 1}.

This is the usual infinite dimensionalsphere used in algebraic topology, the topological inductive limit of Sn ⊂Sn+1 ⊂. .

. .

ASPECTS OF THE THEORY OF INFINITE DIMENSIONAL MANIFOLDS13We show that S∞is a smooth manifold by describing an explicit smooth atlas forS∞, the stereographic atlas. Choose a ∈S∞(”south pole”).

LetU+ := S∞\ {a},u+ : U+ →{a}⊥,u+(x) = x−⟨x,a⟩a1−⟨x,a⟩,U−:= S∞\ {−a},u−: U−→{a}⊥,u−(x) = x−⟨x,a⟩a1+⟨x,a⟩.From an obvious drawing in the 2-plane through 0, x, and a it is easily seen that u+ isthe usual stereographic projection. We also getu−1+ (y) = |y|2−1|y|2+1a +2|y|2+1yfor y ∈{a}⊥\ {0}and (u−◦u−1+ )(y) =y|y|2 .

The latter equation can directly be seen from the drawingusing ”Strahlensatz”.The two stereographic charts above can be extended to charts on open sets in R∞insuch a way that S∞becomes a splitting submanifold of R∞:˜u+ : R∞\ [0, +∞)a →a⊥+ (−1, +∞)a˜u+(z) := u+( z|z|) + (|z| −1)a.Since the model space R∞of S∞has the bornological approximation property by4.9, and is reflexive, by 4.10 the operational tangent bundle of S∞equals the kinematicone: DS∞= TS∞.We claim that TS∞is diffeomorphic to {(x, v) ∈S∞× R∞: ⟨x, v⟩= 0}.The Xx ∈TxS∞are exactly of the form c′(0) for a smooth curve c : R →S∞with c(0) = x by 4.11.Then 0 =ddt|0⟨c(t), c(t)⟩= 2⟨x, Xx⟩.For v ∈x⊥we usec(t) = cos(|v|t)x + sin(|v|t) v|v|.The construction of S∞works for any positive definite bounded bilinear form on anyconvenient vector space.6.3. Example.

The Grassmannians and the Stiefel manifolds.The Grassmann manifold G(k, ∞; R) is the set of all k-dimensional linear subspaces ofthe space of all finite sequences R∞.The Stiefel manifold O(k, ∞; R) of orthonormal k-frames is the set of all linear isome-tries Rk →R∞, where the latter space is again equipped with the standard weak innerproduct described at the beginning of 6.2.The Stiefel manifold GL(k, ∞; R) of all k-frames is the set of all injective linearmappings Rk →R∞.There is a canonical transposition mapping ()t : L(Rk, R∞) →L(R∞, Rk) which isgiven byAt : R∞incl−−→RN = (R∞)′A′−→(Rk)′ = Rkand satisfies ⟨At(x), y⟩= ⟨x, A(y)⟩. The transposition mapping is bounded and linear,so it is real analytic.Then we haveGL(k, ∞) = {A ∈L(Rk, R∞) : At ◦A ∈GL(k)},

14ANDREAS KRIEGL PETER W. MICHORsince At ◦A ∈GL(k) if and only if ⟨Ax, Ay⟩= ⟨AtAx, y⟩= 0 for all y implies x = 0,which is equivalent to A injective. So in particular GL(k, ∞) is open in L(Rk, R∞).The Lie group GL(k) acts freely from the right on the space GL(k, ∞).

Two elementsof GL(k, ∞) lie in the same orbit if and only if they have the same image in R∞. Wehave a surjective mapping π : GL(k, ∞) →G(k, ∞), given by π(A) = A(Rk), where theinverse images of points are exactly the GL(k)-orbits.Similarly we haveO(k, ∞) = {A ∈L(Rk, R∞) : At ◦A = Idk}.Now the Lie group O(k) of all isometries of Rk acts freely from the right on the spaceO(k, ∞).

Two elements of O(k, ∞) lie in the same orbit if and only if they have thesame image in R∞. The projection π : GL(k, ∞) →G(k, ∞) restricts to a surjectivemapping π : O(k, ∞) →G(k, ∞) and the inverse images of points are now exactly theO(k)-orbits.6.4.

Lemma (Iwasawa decomposition). Let T(k; R) be the group of all upper tri-angular k × k-matrices with positive entries on the main diagonal.

Then each B ∈GL(k, ∞) can be written in the form B = p(B)◦q(B), with unique p(B) ∈O(k, ∞) andq(B) ∈T(k). The mapping q : GL(k, ∞) →T(k) is real analytic.6.5.

Theorem. The following are a real analytic principal fiber bundles:(O(k, ∞; R), π, G(k, ∞; R), O(k, R)),(GL(k, ∞; R), π, G(k, ∞; R), GL(k, R)),(GL(k, ∞; R), p, O(k, ∞; R), T(k; R)).The last one is trivial.The embeddings Rn →R∞induce real analytic embeddings, which respect the prin-cipal right actions of all the structure groupsO(k, n) →O(k, ∞),GL(k, n) →GL(k, ∞),G(k, n) →G(k, ∞).All these cones are inductive limits in the category of real analytic (and smooth) mani-folds.6.6.Theorem.

The following manifolds are real analytically diffeomorphic to thehomogeneous spaces indicated:GL(k, ∞) ∼= GL(∞)IdkL(Rk, R∞−k)0GL(∞−k)O(k, ∞) ∼= O(∞)/Idk × O(∞−k)G(k, ∞) ∼= O(∞)/O(k) × O(∞−k).The universal vector bundle (E(k, ∞), π, G(k, ∞), Rk) is defined as the associated bundleE(k, ∞) = O(k, ∞)[Rk]= {(Q, x) : x ∈Q} ⊂G(k, ∞) × R∞.The tangent bundle of the Grassmannian is then given byTG(k, ∞) = L(E(k, ∞), E(k, ∞)⊥).

ASPECTS OF THE THEORY OF INFINITE DIMENSIONAL MANIFOLDS156.7 Theorem. The principal bundle (O(k, ∞), π, G(k, ∞)) is classifying for finite di-mensional principal O(k)-bundles and carries a universal real analytic O(k)-connectionω ∈Ω1(O(k, ∞), o(k)).This means: For each finite dimensional smooth or real analytic principal O(k)-bundle P →M with principal connection ωP there is a smooth or real analytic mappingf : M →G(k, ∞) such that the pullback O(k)-bundle f ∗O(k, ∞) is isomorphic to Pand the pullback connection f ∗ω = ωP via this isomorphism.6.8.

The Lie group GL(∞, R). The canonical embeddings Rn →Rn+1 onto the firstn coordinates induce injections GL(n) →GL(n + 1).

The inductive limit isGL(∞) := lim−→nGL(n)in the category of sets. Since each GL(n) also injects into L(R∞, R∞) we can visualizeGL(∞) as the set of all N ×N-matrices which are invertible and differ from the identityin finitely many entries only.We also consider the Lie algebra gl(∞) of all N × N-matrices with only finitely manynonzero entries, which is isomorphic to R(N×N), and we equip it with this convenientvector space structure.

Then gl(∞) = lim−→n gl(n) in the category of real analytic map-pings.Claim. gl(∞) = L(RN, R(N)) as convenient vector spaces.

Composition is a boundedbilinear mapping on gl(∞).6.9. Theorem.

GL(∞) is a real analytic Lie group modeled on R∞, with Lie algebragl(∞) and is the inductive limit of the Lie groups GL(n) in the category of real ana-lytic manifolds. The exponential mapping is well defined, is real analytic and a localreal analytic diffeomorphism onto a neighborhood of the identity.

The Campbell-Baker-Hausdorffformula gives a real analytic mapping near 0 and expresses the multiplicationon GL(∞) via exp. The determinant det : GL(∞) →R \ 0 is a real analytic homomor-phism.

We have a real analytic left action GL(∞) × R∞→R∞, such that R∞\ 0 isone orbit, but the injection GL(∞) ֒→L(R∞, R∞) does not generate the topology.Proof. Since the exponential mappings are compatible with the inductive limits all theseassertions follow from the inductive limit property.□6.10.

Theorem. Let g be a Lie subalgebra of gl(∞).

Then there is a smoothly arcwiseconnected splitting Lie subgroup G of GL(∞) whose Lie algebra is g. The exponentialmapping of GL(∞) restricts to that of G, which is local diffeomorphism near zero. TheCampbell Baker Hausdorffformula gives a real analytic mapping near 0 and has theusual properties, also on G.Proof.

Let gn := g ∩gl(n), a finite dimensional Lie subalgebra of g.Then S gn =g. The convenient structure g = lim−→n gn coincides with the structure inherited as acomplemented subspace, since gl(∞) carries the finest locally convex structure.So for each n there is a connected Lie subgroup Gn ⊂GL(n) with Lie algebra gn.Since gn ⊂gn+1 we have Gn ⊂Gn+1 and we may consider G := Sn Gn ⊂GL(∞).Each g ∈G lies in some Gn and may be connected to Id via a smooth curve there,which is also smooth curve in G, so G is smoothly arcwise connected.All mappings exp |gn : gn →Gn are local real analytic diffeomorphisms near 0,so exp : g →G is also a local real analytic diffeomorphism near zero onto an openneighborhood of the identity in G. The rest is clear.□

16ANDREAS KRIEGL PETER W. MICHOR6.11. Examples.The Lie group SO(∞, R) is the inductive limitSO(∞) := lim−→nSO(n) ⊂GL(∞).It is the connected Lie subgroup of GL(∞) with the Lie algebra o(∞) = {X ∈gl(∞) :Xt = −X} of skew elements.

Obviously we haveSO(∞) = {A ∈GL(∞) : ⟨Ax, Ay⟩= ⟨x, y⟩for all x, y ∈R∞and det(A) = 1}.The Lie group O(∞) is the inductive limitO(∞) := lim−→nO(n) ⊂GL(∞)= {A ∈GL(∞) : ⟨Ax, Ay⟩= ⟨x, y⟩for all x, y ∈R∞}.It has two connected components, that of the identity is SO(∞).The Lie group SL(∞) is the inductive limitSL(∞) : = lim−→nSL(n) ⊂GL(∞)= {A ∈GL(∞) : det(A) = 1}.It is the connected Lie subgroup with Lie algebra sl(∞) = {X ∈gl(∞) : Trace(X) = 0}.6.12. We stop here to give examples.Of course this method is also applicable forthe complex versions of the most important homogeneous spaces.

This will be treatedelsewhere.ReferencesAbraham, Ralph, Lectures of Smale on differential topology, Lecture Notes, Columbia University, NewYork, 1962.Boman, Jan, Differentiability of a function and of its compositions with functions of one variable,Math. Scand.

20 (1967), 249–268.Cervera, Vicente; Mascaro, Francisca; Michor, Peter W., The orbit structure of the action of thediffeomorphism group on the space of immersions, Preprint (1989).Eells, James, A setting for global analysis, Bull AMS 72 (1966), 571–807.Fantappi´e, L., I functionali analitici, Atti Accad. Naz.

Lincei Mem. 3-11 (1930), 453–683.Fantappi´e, L., ¨Uberblick ’uber die Theorie der analytischen Funktionale und ihre Anwendungen, Jahres-ber.

Deutsch. Mathem.

Verein. 43 (1933), 1–25.Fr¨olicher, Alfred; Kriegl, Andreas, Linear spaces and differentiation theory, Pure and Applied Mathe-matics, J. Wiley, Chichester, 1988.Gil-Medrano, Olga; Peter W. Michor, The Riemannian manifold of all Riemannian metrics, Preprint(1989).Grabowski, Janusz, Free subgroups of diffeomorphism groups, Fundamenta Math.

131 (1988), 103–121.Grauert, Hans, On Levi’s problem and the embedding of real analytic manifolds, Annals of Math. 68(1958), 460–472.Jarchow, Hans, Locally convex spaces, Teubner, Stuttgart, 1981.Keller, H. H., Differential calculus in locally convex spaces, vol.

417, Springer Lecture Notes, 1974.

ASPECTS OF THE THEORY OF INFINITE DIMENSIONAL MANIFOLDS17Kriegl, Andreas, Die richtigen R¨aume f¨ur Analysis im Unendlich - Dimensionalen, Monatshefte Math.94 (1982), 109–124.Kriegl, Andreas, Eine kartesisch abgeschlossene Kategorie glatter Abbildungen zwischen beliebigenlokalkonvexen Vektorr¨aumen, Monatshefte f¨ur Math. 95 (1983), 287–309.Kriegl, Andreas, Some remarks on germs in infinite dimensions, Preprint (1989).Kriegl, Andreas; Nel, Louis D., A convenient setting for holomorphy, Cahiers Top.

G´eo. Diff.

26 (1985),273–309.Kriegl, Andreas; Michor, Peter W., A convenient setting for real analytic mappings, 52 p., to appear,Acta Mathematica (1990).Kriegl, Andreas; Michor, Peter W., Foundations of Global Analysis, in the early stages of preparation.Leslie, Joshua, On the group of real analytic diffeomorphisms of a compact real analytic manifold,Transact. Amer.

Math. Soc.

274 (2) (1982), 651–669.Michor, Peter W., Manifolds of differentiable mappings, Shiva, Orpington, 1980.Michor, Peter W., Manifolds of smooth mappings IV: Theorem of De Rham, Cahiers Top. Geo.

Diff.24 (1983), 57–86.Michor, Peter W., Gauge theory for the diffeomorphism group, Proc. Conf.

Differential GeometricMethods in Theoretical Physics, Como 1987, K. Bleuler and M. Werner (eds), Kluwer, Dordrecht,1988, pp. 345–371.Michor, Peter W., The moment mapping for unitary representations, Preprint 1989.Milnor, John, Remarks on infinite dimensional Lie groups, Relativity, Groups, and Topology II, LesHouches, 1983, B.S.

DeWitt, R. Stora, Eds., Elsevier, Amsterdam, 1984.Pressley, Andrew; Segal, Graeme, Loop groups, Oxford Mathematical Monographs, Oxford UniversityPress, 1986.Institut f¨ur Mathematik, Universit¨at Wien, Strudlhofgasse 4, A-1090 Wien, Austria.

---

출처: arXiv:9202.206 • 원문 보기