# Questions tagged [hilbert-manifolds]

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

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### $\sigma$-product of the Hilbert cube

Given a homogeneous space $X$ and $p\in X$, we define the sigma product to be the following subspace of $X^\omega$: $$\sigma X=\{\mathbf x \in X^\omega:x_n=p\text{ eventually}\}$$
("eventually&...

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### 'Closedness' for infinite-dimensional separable Hilbert manifolds

In the following content, the Hilbert space refers to the unique separable infinite-dimensional real Hilbert space $H$. Hilbert manifolds are locally modelled on this space (so they must be infinite-...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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(...

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### 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

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
...

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### 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 ...

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### 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 ...

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### 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 $\prod_{n=1}^{\infty} (0,1)$ (...

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### 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 ...

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### 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}
...

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### 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?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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-...