Questions tagged [hilbert-manifolds]

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

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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 $\prod_{n=1}^{\infty} (0,1)$ (...
8
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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 ...
5
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156 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 ...
4
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121 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(...
3
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0answers
74 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 ...
3
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0answers
84 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
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0answers
160 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 ...
2
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75 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} ...
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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