Questions tagged [infinite-dimensional-manifolds]
The infinite-dimensional-manifolds tag has no usage guidance.
142
questions
2
votes
1
answer
182
views
$\sigma$-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&...
CommunityBot
- 1
2
votes
0
answers
62
views
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 ...
- 5,011
1
vote
0
answers
151
views
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.
...
- 4,353
1
vote
1
answer
131
views
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 ...
- 2,329
5
votes
2
answers
609
views
Image of the Hilbert space under a continuous bijection
Consider a continuous bijection $X\to Y$ such that $X$ is homeomorphic to a separable infinite dimensional Hilbert space. I wonder what can be said about topological properties of $Y$.
To exclude ...
- 28.5k
0
votes
0
answers
144
views
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", ...
- 5,632
4
votes
1
answer
211
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 ...
- 17.2k
3
votes
0
answers
89
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 ...
- 61
8
votes
1
answer
320
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 ...
- 178k
3
votes
0
answers
105
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 ...
- 987
2
votes
0
answers
125
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 ...
- 4,776
5
votes
1
answer
232
views
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^{\...
- 20.7k
11
votes
5
answers
2k
views
Ricci Curvature in infinite dimensions?
Is there a good notion of "Ricci curvature" in infinite dimensions?
My intuitive understanding of Ricci curvature is that it is some kind of an "average" of the curvature tensor over "different ...
- 60.2k
8
votes
1
answer
219
views
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 ...
- 3,390
38
votes
1
answer
1k
views
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 ...
- 11.4k
3
votes
0
answers
369
views
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
votes
0
answers
88
views
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 ...
- 4,353
5
votes
0
answers
149
views
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 ...
- 334
1
vote
0
answers
103
views
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 ...
- 60.2k
1
vote
1
answer
145
views
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 ...
- 60.2k
8
votes
1
answer
424
views
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 ...
- 60.2k
12
votes
1
answer
510
views
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 ...
- 20.7k
5
votes
1
answer
450
views
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 ...
- 9,539
2
votes
1
answer
110
views
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_{...
- 866
6
votes
1
answer
201
views
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{...
- 11.4k
4
votes
3
answers
550
views
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 \...
- 39
3
votes
0
answers
83
views
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 ...
- 1,253
18
votes
2
answers
757
views
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 $E \rightarrow H$ ...
- 546
7
votes
0
answers
129
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 ...
- 11.4k
3
votes
1
answer
236
views
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/...
- 11.4k
1
vote
0
answers
42
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 ...
- 60.2k
2
votes
2
answers
394
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 ...
- 9,749
3
votes
0
answers
144
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 $D^n$?
- 7,373
6
votes
1
answer
189
views
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
- 3,049
4
votes
0
answers
182
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 ...
2
votes
0
answers
106
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 ...
6
votes
1
answer
460
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 ...
- 7,373
4
votes
0
answers
204
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(...
- 312
1
vote
0
answers
92
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
- 312
2
votes
0
answers
124
views
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 ...
- 312
2
votes
1
answer
151
views
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 ...
- 25.6k
5
votes
1
answer
364
views
Manifolds of maps
Suppose $M$ is a compact $n$-dimensional manifold without boundary. Let $H^s(M,M)$ denote the Sobolev space on $M$, defined as all maps from $M$ to $M$ whose distributional derivatives up to order $s$ ...
- 60.2k
2
votes
0
answers
130
views
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,\...
- 518
18
votes
0
answers
537
views
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)...
- 7,373
3
votes
0
answers
187
views
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 ...
- 1,675
3
votes
1
answer
123
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 ...
- 5,472
2
votes
0
answers
183
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 ...
- 667
5
votes
1
answer
178
views
$c^\infty$ topology on $L(E, F)$
In Kriegl/Michor's "Convenient Setting for Global Analysis", they put on the set $L(E, F)$ of bounded linear operators between locally convex spaces $E$, $F$ the subspace topology induced by the ...
- 2,063
10
votes
1
answer
372
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 $\Sigma$...
- 28.8k
1
vote
0
answers
41
views
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 ...
- 183