# Questions tagged [hilbert-manifolds]

A Hilbert manifold is a manifold based on a Hilbert space.

23
questions

1
vote

0
answers

70
views

### Exponential map on the Hilbert manifold $\Omega(M,q_0,q_1)$

Consider $M$ to be a compact manifold and consider the based loop space $\Omega(M,q_0,q_1)$ of loops of class $W^{1,2}$. It can be shown that that this has a structure of an hilbert manifold. It's ...

10
votes

2
answers

383
views

### Implicit function theorem with continuous dependence on parameter

Let $X,Y$ be Hilbert spaces and $P$ a topological space$^1$ and $p_0\in P$.
Let $f:X\times P\to Y$ be a continuous map such that
for any parameter $p\in P$, $f_p:= f|_{X\times \{p\}}:X\to Y$ is ...

3
votes

1
answer

127
views

### Tangent space of smooth Hilbert submanifolds

Let $X, Y$ be Hilbert spaces and $F:X \rightarrow Y$ smooth. Assume that $M := F^{-1}(0) \subset X$ is a smooth submanifold. Is it true that for any $x\in M$, the tangent space $T_xM$ is a Hilbert ...

4
votes

0
answers

108
views

### Almost-geodesics on a Riemannian Hilbert manifold which are still almost geodesics in some submanifold

Let $H$ be a separable infinite dimensional Hilbert space, and consider it as a Hilbert manifold in the usual way (that is, with the single chart with the identity map). It is known that there always ...

4
votes

0
answers

160
views

### Fredholm transversality

Let $M,N$ be two Banach or Hilbert manifolds. Assume that $P\subset N$ is a smooth submanifold of $N$.
We say that a smooth map $f:M \to N$ is Fredholm transversal to $P$ if for every $x\in M$ with $f(...

1
vote

0
answers

87
views

### Infinite dimensional smooth projective geometry

Are there two infinite dimensional (Banach or Hilbert) manifolds $(P,L)$ which satisfy the axioms of a smooth projective geometry desribed in this page: Smooth Projectove Geometry

3
votes

0
answers

116
views

### What is a $C^\infty$ diffeomorphism from $\ell_2\setminus\{0\}$ to $\ell_2$ which is the identity outside a ball?

Let $\ell_2:=\{x=(x_n)_{n\in\mathbb N}:\ \|x\|^2:=\sum_n|x_n|^2<\infty\}$ with its natural norm. According to Wikipedia https://en.wikipedia.org/wiki/Kuiper%27s_theorem and to other sources, it is ...

3
votes

0
answers

172
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 ...

1
vote

1
answer

174
views

### Can a continuous map on a Hilbert manifold be approximated by a map which has infinitely many critical points?

It is well-known that a continuous map $f:M\to\mathbb{R}^n$ from a Hilbert manifold can be closely approximated by a smooth map $g:M\to\mathbb{R}^n$ which has no critical points.
But, can such a ...

3
votes

1
answer

119
views

### Does their exist something like L^2 Mapping spaces to general manifolds?

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 ...

1
vote

1
answer

77
views

### What are examples of a second order operational tangent vector on an infinite dimensional Hilbert space

In the book "a convenient setting for global analysis" they describe the order of an operational tangent vector on a convenient vector space.
http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf
...

8
votes

1
answer

396
views

### Smooth trivialization of smooth Hilbert bundles

In page 67 of Topology and Analysis by Booss and Bleecker, it is claimed that any Hilbert bundle is topologically trivial. Clearly, any smooth Hilbert bundle over a smooth manifold is topologically ...

7
votes

1
answer

369
views

### Are separable F-spaces (completely metrizable topological vector space) homeomorphic to $l_2$?

An F-space is a completely metrizable topological vector space, i.e. the vector topology is induced by a complete metric. A Fréchet space is, by definition, a locally convex F-space.
It is known that ...

14
votes

0
answers

240
views

### Spaces locally modelled on $L^2(\mathbb R)$

In this recent question, I learned that any two separable Banach spaces are homeomorphic. Based on some readings, I'm guessing that $L^2(\mathbb R)$ is homeomorphic to $\prod_{n=1}^{\infty} (0,1)$ (...

5
votes

1
answer

338
views

### Reference: Stochastic Analysis on Hilbert Manifolds

I'm looking for a reference to a book which develops an It\^{o} lemma for semi-martingales with values in infinite dimensional Hilbert-Manifolds. I expect the techniques to be the same but still I ...

2
votes

0
answers

77
views

### Christoffel symbols of a moduli of smooth curves

The Setting:
Let $H$ be the Hilbert space of all class $C^k$-curves into $\mathbb{R}$ with inner product: \begin{equation}
<f,g>:=\int_{\mathbb{R}} f'(x)g'(x) e^{-x}dx
\end{equation}
...

0
votes

1
answer

84
views

### Embedding Riemmanian Manifold Linearly

Given a Hilbert Manifold $M$ does there exist a smooth map into some very large Hilbert space taking geodesics to straight lines?

5
votes

0
answers

162
views

### are smooth homotopic open embedding of Hilbert manifolds smoothly isotopic?

A theorem of Chapman (Chapman, T. A. Homotopic homeomorphisms of infinite-dimensional manifolds. Compositio Math. 27 (1973), 135–140) states that every two open embeddings $f,g:M\to N$ of Hilbert ...

8
votes

0
answers

401
views

### About Palais' remark that an isometry of Riemannian manifolds does not induce an isometry of the Hilbert manifolds of curves

In the paper ``Morse theory on Hilbert manifolds'' (1963), on page
326, Richard Palais makes a remark that if $\phi\colon V \to W$ is an
isometry (of submanifolds of $\mathbb{R}^n$), then this does ...

4
votes

1
answer

226
views

### Hilbert manifolds and embedding

In the Wikipedia article on Hilbert manifolds, it is claimed that every Hilbert manifold can be smoothly embedded onto an open subset of the model Hilbert space. However, no explicit reference is ...

34
votes

4
answers

5k
views

### How are infinite-dimensional manifolds most commonly treated?

I originally posted this question on StackExchange, where it was suggested I post here. It was also suggested I read about Hilbert manifolds and Fréchet manifolds. Nevertheless, I am still looking for ...

3
votes

2
answers

1k
views

### Infinite dimensional manifold

In Hamiltonian mechanics, one essentially work with $\mathbb{R}^{2n}$. However, this is only a local description of our configuration manifold $M$. More precisely, the mechanical system is regarded as ...

3
votes

2
answers

617
views

### Gluing in Morse homology for Hilbert manifolds

Is there any reference for gluing in the context of Morse homology on Hilbert manifolds?
Gluing is pretty standard in Morse homology for finite-dimensional manifolds. Unfortunately, in the infinite-...