# Questions tagged [infinite-dimensional-manifolds]

The infinite-dimensional-manifolds tag has no usage guidance.

143
questions

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### The closure of the space of Riemannian metrics with a fixed isometry class

Let $M$ be a closed manifold, and let $\mathscr{M}$ be the space of all Riemannian metrics over $M$. It is known that this is a Fréchet manifold. Consider also $\mathscr{D}$ the diffeomorphisms group ...

2
votes

0
answers

65
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### How far can one get by counting spaces of solutions this way?

I am quite used to "counting"/computing finite dimensions. For example, one would expect a hypersurface in $\mathbb{C}^3$ to have dimension $3 - 1 = 2$. But it is often the case that the ...

1
vote

0
answers

188
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### General linear group in infinite dimensions

Let $V$ be a vector space over the field $k$. Upon assuming the Axiom of Choice, we know that $V$ has a well-defined dimension $N$, and hence a well-defined basis $B$. Suppose that $N$ is not finite.
...

1
vote

1
answer

145
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### Commuting time dependent vector fields and pullback invariance

Let $X_t, Y_t \in C^\infty(\mathbb{R}; \mathfrak{X}^\infty(M))$ be (smooth, or something else if it's necessary) time dependent vector fields.
Is there some analogue of the following fact in finite ...

2
votes

1
answer

201
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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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156
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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", ...

4
votes

1
answer

213
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### 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 ...

3
votes

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answers

98
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### 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 ...

8
votes

1
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327
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### 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 ...

3
votes

0
answers

107
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### "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 ...

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128
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### 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 ...

5
votes

1
answer

242
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### Space of spacelike embeddings as infinite-dimensional manifold

Consider a four-dimensional Lorentzian manifold $(\mathcal{M},g)$ and a $3$-dimensional compact manifold $\Sigma$, such that there exists a spacelike embedding $i:\Sigma\to\mathcal{M}$ so that $h:=i^{\...

8
votes

1
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225
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### Is the Borel lemma projection a smooth principal bundle?

Consider the Fréchet spaces $C^\infty(\mathbb{R},\mathbb{R})$ and $\mathbb{R}^\infty$, and the continuous linear map
$$
J\colon C^\infty(\mathbb{R},\mathbb{R}) \to \mathbb{R}^\infty
$$
returning the ...

3
votes

0
answers

398
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### What's the problem with the evaluation map not being continuous?

When introducing differentiable functions between locally convex spaces, many authors (e.g. Bastiani, Keller, Kriegl-Michor) notice that the evaluation map
$$ E\times E^*\to\mathbb R,\qquad (x,L)\...

2
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92
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### Nonlinear automorphisms of projective spaces and the axiom of choice

Let $k$ be a field and $\mathbf{P}$ a projective space over $k$. If we accept the axiom of choice (AC), then $\mathbf{P}$ has a basis and a dimension $m$, and if $m$ is finite, the automorphism group ...

5
votes

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153
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### Is the Grassmannian of a Banach space an infinite dimensional manifold?

Grassmannian of complemented subspaces in a Banach space is a Banach manifold. This is explained for example in the thesis of Douady and is rather analogous to the finite-dimensional case.
I would ...

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0
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103
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### Infinite-dimensional Lie group corresponding to $U\mathfrak{g}$?

Let $\mathfrak{g}$ be a Lie algebra. The universal enveloping algebra $U\mathfrak{g}$ is then an infinite-dimensional associative algebra which can be endowed with the structure of a Lie algebra. Is ...

1
vote

1
answer

161
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### Charts for the Banach manifold of smooth almost complex structures $\mathcal{J}^{l}$

Consider the closure in the $C^l$-topology of the space of smooth almost complex structrues of a symplectic manifold $(M,\omega)$. We will denote this space by $\mathcal{J}^l$.
It's a very used fact ...

8
votes

1
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438
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### Topological group locally homeomorphic to the Hilbert cube

Does there exist a topological group which is locally homeomorphic to the Hilbert cube $[0,1]^{\mathbb N}$?
Let me note that Hilbert cube has the fixed point property and thus it is not homeomorphic ...

12
votes

1
answer

539
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### Different smooth structures on the infinite jet bundle (for the purposes of calculus of variations)

Let $\pi:Y\rightarrow X$ be a (smooth, finite dimensional) fibred manifold. Since no other fibrations will be considered on $Y$, I will identify $(Y,\pi,X)$ with $Y$. The finite order jet bundles are ...

38
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1
answer

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### Sequences with 0's in $\mathbb R ^\omega$

Let $\mathbb R ^\omega$ be the set of all sequences of real numbers in the product topology.
Let $X$ be the set of all sequences in $\mathbb R ^\omega$ which have at least one 0.
Let $Y$ be the set of ...

2
votes

1
answer

113
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### Smooth dependence in the fixed point theorem between complete Fréchet manifolds

Let $X,Y$ be complete metric spaces, and let $\Sigma:X\times Y\rightarrow Y$ be a continous mapping which satisfies the following property: there exists a $C<1$, such that for all $x\in X$ and $y_{...

5
votes

1
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472
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### Which completion of the configuration space of $n$ distinct points in $\mathbb{R}^d$ is better suited for numerical analysis?

(My original post starts here, and ends right before the Edit part. I am keeping it so that the comments and answer make sense, but what I am really interested in is what is in the Edit section.)
My ...

6
votes

1
answer

208
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### On the orbit of a Fréchet Lie group action

Suppose that $G$ is a Fréchet Lie group acting on a Fréchet manifold $X$.
Fix $x\in X$ and let $\alpha(t)$ be a smooth path in $X$ such that
$$
\begin{cases}
\alpha(0)=x\\
\alpha(t)\in G\cdot x.
\end{...

3
votes

0
answers

85
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### Couniversality of Lie integration in different categories of manifolds/smooth spaces

A fairly reasonable interpretation of Lie II and Lie III seems to be that the category of Lie algebras is a coreflective subcategory of the category of Lie groups, so that the Lie group integrating a ...

4
votes

3
answers

574
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### Intersection modulo 2 theory for infinite dimensional manifolds?

For finite dimensional manifolds, there is a lot of theory about when the number of intersections (modulo $2$) of certain objects are preserved under homotopy. I'll give two quick examples:
Let $f:X \...

18
votes

2
answers

768
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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 $E \rightarrow H$ ...

7
votes

0
answers

132
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### 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 ...

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0
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42
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### 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 ...

2
votes

2
answers

423
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### 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 ...

3
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147
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### 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 $D^n$?

4
votes

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189
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### 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 ...

2
votes

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answers

106
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### 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 ...

6
votes

1
answer

496
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### 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 ...

4
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0
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224
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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(...

1
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0
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92
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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

6
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1
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194
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### A smooth map on a Banach manifold whose pointwise rank is finite but its rank is not globally bounded

Is there a connected Banach manifold $M$ and a smooth map $f:M \to M$ such that the rank of $Df_x$ is finite for every $x\in M$ but this rank is not uniformly bounded

3
votes

1
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248
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### Finite-dimensional argument for Morse-Smale pairs?

Using the Sard-Smale theorem, it is relatively easy to show that Morse-Smale pairs on a manifold $M$ (i.e. pairs $(f,g)$ where $g$ is a metric on $M$, $f$ is a Morse function on $M$, and the stable/...

2
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0
answers

129
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### Smooth derivations of a Banach space

Let $E$ be a real (or complex) Banach space. By $C^\infty(E) $ we mean the space of all functions $f:E\to \mathbb{R}(f:E\to \mathbb{C})$ which are smooth in the sense of Frechet diffetentiability. A ...

2
votes

1
answer

157
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### What is the manifold structure of smooth path on infinite dimensional manifolds?

What is the manifold structure of smooth path on infinite dimensional manifolds?
In the the paper "manifolds of smooth maps by Michor", it was mentioned that it is possible to put manifold structure ...

2
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0
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132
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### Differential of the Rabinowitz Action Functional

On an exact Hamiltonian system $(M,d\alpha,H)$ define the Rabinowitz action functional
$$\mathcal{A}^H \colon C^\infty(\mathbb{S}^1,M) \times (0,+\infty) \to \mathbb{R}$$
by
$$\mathcal{A}^H(\gamma,\...

18
votes

0
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539
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### A curious switch in infinite dimensions

Let $V$ be a finite dimensional real vector space. Let $GL(V)$ be the set of invertible linear transformations, and $\Phi(V)$ be the set of all linear transformations. We can also characterize $\Phi(V)...

3
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0
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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 ...

3
votes

1
answer

129
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### 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 ...

2
votes

0
answers

194
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### 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 ...

10
votes

1
answer

374
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### 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
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0
answers

43
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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 ...

15
votes

1
answer

580
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### 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 ...

3
votes

0
answers

237
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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 ...

5
votes

1
answer

676
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### 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 ...