# Questions tagged [infinite-dimensional-manifolds]

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### Contractibility of infinite dimensional spheres and some other infinite dimensional manifolds

It is known that spheres in Banach spaces are contractible according to Yoav Benyamini, Yaki Sternfeld, "Spheres in infinite-dimensional normed spaces are Lipschitz contractible", ...
190 views

### In what topology does Gromov's lemma hold on noncompact symplectic manifolds?

In symplectic geometry, it is commonly said that the set of almost complex structures tamed to a symplectic form is contractible'' even on noncompact symplectic manifolds. In my understanding one ...
54 views

### How to calculate the exterior derivative on manifolds of smooth mappings?

Let $S$ be a compact finite-dimensional manifold $S$ and $(M, \omega)$ a symplectic manifold. The space of smooth maps from $S$ to $M$, denoted by $\mathcal{M}$, has a canonical infinite-dimensional ...
295 views

### Generalized functions in infinite dimensions

What theories are there for generalized functions (distributions) in infinite dimensions? In particular, suppose your "infinite dimensional manifold" is $\mathfrak{M}:=C^\infty(S^1)$. Is ...
97 views

### "Practical" references on mapping spaces as infinite-dimensional manifolds

I am studying spaces of the form $C^{k}(\mathcal{M},\mathcal{N})$ between manifolds ($k=\infty$ allowed) and I am looking for extensive references, especially analysing their topology and smooth ...
114 views

### What are examples of infinite-dimensional Banach spaces that are also measure spaces?

I am interested in examples of infinite-dimensional vector spaces that are Banach spaces or even Hilbert spaces are measure spaces Instead of the full vector space, subsets with measure structure ...
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### What is known about the "unitary group" of a rigged Hilbert space?

Suppose that $(E,H)$ is a rigged (infinite dimensional, separable) Hilbert space, i.e. $H$ is a Hilbert space, and $E$ is a Fréchet space, equipped with a continuous linear injection $E \rightarrow H$ ...
115 views

### Is the symplectomorphism group of a compact manifold a tame Fréchet Lie subgroup of $\operatorname{Diff}(X)$?

In the famous paper Hamilton, Richard S. The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222, Hamilton introduced the category of tame Fréchet Lie ...
1 vote
41 views

### Continuous surjection from $X(D_n)$ onto $\operatorname{Homeo}_0(D_n)$

Let $n>1$ and let $\mathfrak{X}(D_n)$ denote the set of continuous vector fields on the closed disc $D_n\subseteq \mathbb{R}^n$. Let $\operatorname{Homeo}_0(D_n)$ be the set of homeomorphism of ...
357 views

### Reconciling some result about the exponential map, the Chow-Rashevskii theorem, and $\mathrm{Diff}_0(M)$

Let $M$ be a $C^{\infty}$ manifold $C^{\infty}$-diffeomorphic to $\mathbb{R}^d$. I've recently come across some results which I'm trying to reconcile. Let $\mathfrak{X}(M)$ denote the set of ...
127 views

### Diffeomorphisms fixing origin and boundary

Let $D^n$ be a disc in $\mathbb{R}^n$. Is there a known characterization of all the diffeomorphisms of $D^n$ fixing the origin and boundary of $D^n$?
153 views

### Generalizations of Sard-Smale Theorem

Sard-Smale theorem holds for Fredholm maps $f:M\rightarrow B$ between separable Banach manifolds $M,N$. There are some constrains relating the Fredholm index $\operatorname{ind}(f)$ of $f$ to its ...
101 views

### Open embedding of non-separable infinite dimensional manifolds

It is well-known (see here) that separable infinite-dimensional topological Hilbert manifolds can be embedded as open sets of the modeling separable Hilbert space. Using that separable Fréchet (in ...
383 views

### CW structure on infinite-dimensional manifolds

It is well-known (due to this work of Palais, I believe) that Banach manifolds are dominated by countable CW complexes. It then follows (due Whitehead, as indicated by Milnor in this work) that they ...
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### Is there a family of Riemannian manifolds with explicitly solvable geodesics for manifold M?

By this paper https://www.mat.univie.ac.at/~michor/rie-met.pdf for any given manifold $M$, there is a Hilbert manifold (I believe its a Hilbert manifold, if not it is still an infinite dimensional ...
112 views

### Frechet Lie groups and their subgroups

1) Let $G$ be a Fréchet Lie group. Let $H$ be a closed subgroup. Is it always true that the centraliser of $H$ is a Fréchet subgroup of the lie group? 2) Is the closed subgroup theorem valid for ...
152 views

### Definition of integrable representation of Kac-Moody algebra

I have seen several definitions of integrable representation $V$ of Kac-Moody algebra $\mathfrak{g}$ online. Which one is the standard one? Are they actually equivalent? First one is $e_i, f_i$ acts ...
354 views

### Smooth vector fields on a surface modulo diffeomorphisms

Let $\Sigma$ be a two-dimensional connected smooth manifold without boundary. (Feel free to assume it is compact and orientable.) Let $\mathcal{X}(\Sigma)$ denote the smooth vector fields on $\Sigma$...
1 vote
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### Subspace of smooth Foliations inside smooth Distributions - Is it a Frechet submanifold or atleast a Retraction?

Assume M is a closed manifold. Does the set of smooth foliations on M form a Frechet submanifold inside the Frechet manifold consisting of smooth distributions? If not, can the set of smooth ...
460 views

### Is the subgroup $\mathrm{Diff}(M,S)$ of $\mathrm{Diff}(M)$ a Lie subgroup?

Denote by $\mathrm{Diff}(M)$ the Lie group of smooth diffeomorphisms on a compact smooth manifold. Its Lie algebra can be viewed as the Lie algebra $\mathfrak X(M)$ of vector fields on $M$. Now, given ...
175 views

### Structure of a group acting on a Hilbert space

Assume a group $G$ acts faithfully by isometries on a separable infinite dimensional Hilbert space $H$ in such a way that the orbits are closed and the quotient $H/G$ is isometric to a finite ...
543 views

### Smooth structure on the space of sections of a fiber bundle and gauge group

Let $\xi$ be a fiber bundle $F\hookrightarrow E\to B$ (where every space is smooth, T2 and second countable), let $\Gamma(\xi)$ be the space of smooth sections. We can complete $\Gamma(\xi)$ with ...
202 views

### Infinite-dimensional manifold $ΩG$ of loops and the complex projective $n$-space

In a review seminar today, I heard the speaker takes the below for granted, but I have no enough background "Yang-Mills (YM) instantons in 4D can be naturally identified with (i.e. have the same ...
542 views

### How to define (and compute) the Cartan-Killing form of the group of volume-preserving diffeomorphisms?

This question was raised a while ago in a blog post by Terry Tao on the Euler-Arnold equation and he called it "quite tricky". Has anyone in the meantime tried to formulate this question precisely, ...
1 vote
80 views

### "Barrier functions" in function spaces [closed]

In general the idea of a "barrier function" (as in the context of "interior point methods") can possibly be thought of as defining a function corresponding to an open subset of $\mathbb{R}^n$ such ...
Given a compact manifold $C$ and a n-manifold $M$ we mostly work on either $C^{\infty}(C,M)$ seen as a Frechet manifold. or $H^{k}(C,M)$ seen has a Hilbert manifold when $k > n/2$. Although both ...