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Infinite Dimensional Manifolds Infinite dimensional manifolds A series of 4 lectures. Version from 2017/05/28 at 13:20:27 Peter W. Michor University of Vienna, Austria XXV International Fall Workshop on Geometry and Physics Instituto de Estructura de la Materia (CSIC) Madrid, Spain August 29 - September 02, 2016 Based on collaborations with: M. Bauer, M. Bruveris, P. Harms, D. Mumford I Mainly Banach manifolds and why they are not enough. I A short introduction to convenient calculus in infinite dim. I Manifolds of mappings (with compact source) and diffeomorphism groups as convenient manifolds I Manifolds of Sobolev mappings n I A zoo of diffeomorphism groups on R I A diagram of actions of diffeomorphism groups I Riemannian geometries on spaces of immersions and shapes. I Right invariant Riem. geom. on Diffeomorphism groups. I Arnold's formula for geodesics on Lie groups. I Solving the Hunter-Saxton equation. I Riemannian geometries on spaces of Riemannian metrics and pulling them back to diffeomorphism groups. I Approximating Euler's equation of fluid mechanics on the full n diffeomorphism group of R . I Robust Infinite Dimensional Riemannian manifolds, Sobolev Metrics on Diffeomorphism Groups, and the Derived Geometry of Shape Spaces. Mainly Banach manifolds; they are not enough. In these lectures I will try to give an overview on infinite dimensional manifolds with special emphasis on: Lie groups of diffeomorphisms, manifolds of mappings, shape spaces, the manifold of all Riemannian metrics on a manifold, and Riemannian differential geometry on all these. Infinite-dimensional manifolds A smooth manifold modelled on the topologival vector space E is a Hausdorff topological space M together with a family of charts (uα; Uα)α2A, such that S 1. Uα ⊆ M are open sets, α2A Uα = M; 2. uα : Uα ! uα(Uα) ⊆ E are homeomorphisms onto open sets uα(Uα); −1 1 3. uβ ◦ uα : uα(Uα \ Uβ) ! uβ(Uα \ Uβ) are C -smooth. In this definition it does not matter, whether E is finite or infinite-dimensional. In fact, if E is finite-dimensional, then n E = R for some n 2 N and we recover the definition of a finite-dimensional manifold. Choice of a modelling space There are many classes of infinite-dimensional locally convex vector spaces E to choose from. With increasing generality, E can be a 1. Hilbert space; 2. Banach space; 3. Fr´echetspace; 4. convenient locally convex vector space. We will assume basic familiarity with Hilbert and Banach spaces. All topological vector spaces are assumed to be Hausdorff. A Fr´echetspace is a locally convex topological vector space X , whose topology can be induced by a complete, translation-invariant metric, i.e. a metric d : X × X ! R such that d(x + h; y + h) = d(x; y). Alternatively a Fr´echetspace can be characterized as a Hausdorff topological space, whose topology may be induced by a countable family of seminorms k · kn, i.e., finite intersections of the sets fy : kx − ykn < "g with some x, n, " form a basis of the topology, and the topology is complete with respect to this family. The Hilbert sphere Let E be a Hilbert space. Then S = fx 2 E : kxk = 1g is a smooth manifold. We can construct charts on S in the following way: For x0 2 S, define the subspace ? Ex0 fx0g = fy 2 E : hy; x0i = 0g isomorphic to E. Charts : q −1 2 ux0 : x 7! x − hx; x0ix0 ; ux0 : y 7! y + 1 − kyk x0 : Ux0 = fx 2 S : hx; x0i > 0g ! ux0 (Ux0 ) = fy 2 Ex0 : kyk < 1g : If E is infinite-dimensional, then the sphere is not compact. The sphere is smoothly contractible, by the following argument [Ramadas 1982]: We consider the homotopy A : `2 × [0; 1] ! `2 through isometries which is given by A0 = Id and by At (a0; a1; a2;:::) = (a0;:::; an−2; an−1 cos θn(t); an−1 sin θn(t); an cos θn(t); an sin θn(t); an+1 cos θn(t); an+1 sin θn(t);:::) 1 1 π for n+1 ≤ t ≤ n , where θn(t) = '(n((n + 1)t − 1)) 2 for a fixed smooth function ' : R ! R which is 0 on (−∞; 0], grows monotonely to 1 in [0; 1], and equals 1 on [1; 1). The mapping A is smooth since it maps smooth curves to smooth curves (see later: convenient calculus). Then 2 A1=2(a0; a1; a2;:::) = (a0; 0; a1; 0; a2; 0;:::) is in `even, and on the 2 other hand A1(a0; a1; a2;:::) = (0; a0; 0; a1; 0; a2; 0;:::) is in `odd. Now At jS for 0 ≤ t ≤ 1=2 is a smooth isotopy on S between the 1 2 identity and A1=2(S ) ⊂ `even. The latter set is contractible in a chart. One may prove in a simpler way that S1 is contractible with a real analytic homotopy with one corner: roll all coordinates one step to the right and then contract in the stereographic chart opposite to (1; 0;::: ). The method explained above has other uses like the proof of the contractibility of the unitary group on Hilbert space. Theorem (Eells, Elworthy) Any (second countable) Hilbert manifold is smoothly diffeomorphic to an open set in the modelling Hilbert space. The manifold Imm(S 1; Rd ) This is the space of smooth, periodic, immersed curves, 1 d 1 1 d 0 1 Imm(S ; R ) = fc 2 C (S ; R ): c (θ) 6= 0; 8θ 2 S g : 1 1 d The modelling space for the manifold is C (S ; R ), the space of smooth, periodic functions. When we talk about manifolds, we have to specify, what topology we mean. On the space 1 1 d C (S ; R ) of smooth functions we consider the topology of uniform convergence in all derivatives, i.e., (k) (k) fn ! f , lim kfn − f k1 = 0 8k 2 ; n!1 N where kf k1 = supθ2S1 jf (θ)j. A basis of open sets is formed by sets of the form n 1 1 d (k) (k) o M(f ; "; k) = g 2 C (S ; R ): kg − f k1 < " ; with " 2 R>0 and k 2 N . 1 1 d I C (S ; R ) is a reflexive, nuclear, separable Fr´echetspace. 1 d 1 1 d I Imm(S ; R ) is an open subset of C (S ; R ). 1 d The manifold ImmC n (S ; R ) Instead of smooth curves, we could consider curves belonging to some other regularity class. Let n ≥ 1 and 1 d n 1 d 0 1 ImmC n (S ; R ) = fc 2 C (S ; R ): c (θ) 6= 0; 8θ 2 S g ; be the space of C n-immersions. Again we need a topology on the n 1 d space C (S ; R ). In this case (k) kf kn;1 = sup kf k1 ; 0≤k≤n 1 1 d is a norm making (C (S ; R ); k · kn;1) into a Banach space. 1 d n 1 d I For n ≥ 1, ImmC n (S ; R ) is an open subset of C (S ; R ). 1 d Thus ImmC n (S ; R ) is a Banach manifold modeled on the space n 1 d C (S ; R ). There is a connection between the spaces of C n-immersions and the space of smooth immersions. As sets we have 1 d \ 1 d Imm(S ; R ) = ImmC n (S ; R ) : n≥1 However, more is true: we can consider the diagram 1 1 d n 1 d n−1 1 d 1 1 d C (S ; R ) ⊆ · · · ⊆ C (S ; R ) ⊆ C (S ; R ) ⊆ · · · ⊆ C (S ; R ) ; and in the category of topological vector spaces as well as in the the category of locally convex vector spaces C 1(S1; d ) = lim C n(S1; d ) R − R n!1 n 1 d is the projective limit of the spaces C (S ; R ). Calculus in Banach spaces Having chosen a modeling space E, we look back at the definition of a manifold and see that requires chart changes to be smooth maps. To a considerable extent multivariable calculus generalizes without problems from a finite-dimensional Euclidean space to Banach spaces, but not beyond. For example, if X , Y are Banach spaces and f : X ! Y a function, we can define the derivative Df (x) of f at x 2 X to be the linear map A 2 L(X ; Y ), such that kf (x + h) − f (x) − A:hk lim Y = 0 : h!0 khkX I want to emphasize in particular two theorems, that are valid in Banach spaces, but fail for Fr´echetspaces: the existence theorem for ODEs and the inverse function theorem. First, the local existence theorem for ODEs with Lipschitz right hand sides. Theorem Let X be a Banach space, U ⊆ X an open subset and F :(a; b) × U ! X a continuous function, that is Lipschitz continuous in the second variable, i.e., kF (t; x) − F (t; y)kX ≤ Ckx − ykX ; for some C > 0 and all t 2 (a; b), x; y 2 U. Then, given (t0; x0) 2 (a; b) × U, there exists x :(t0 − "; t0 + ") ! X , such that @t x(t) = F (t; x(t)) ; x(t0) = x0 : In fact more can be said: the solution is as regular as the right hand side, if the right hand side depends smoothly on some parameters, then so does the solution and one can estimate the length of the interval of existence; one can also get by with less regularity of F (t; x) in the t-variable. For x 2 X , let Br (x) = fy 2 X : ky − xkX < rg be the open r-ball.
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