Questions tagged [infinite-dimensional-manifolds]

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6
votes
1answer
127 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{...
3
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0answers
64 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 ...
4
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3answers
389 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 \...
18
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2answers
626 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 $E \rightarrow H$ ...
6
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0answers
82 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 ...
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0answers
39 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 ...
2
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2answers
239 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 ...
3
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0answers
88 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 $D^n$?
4
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0answers
124 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
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0answers
94 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
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1answer
211 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 ...
4
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0answers
118 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(...
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0answers
78 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
6
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1answer
160 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
votes
1answer
131 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/...
2
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0answers
81 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 ...
1
vote
1answer
104 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 ...
2
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0answers
75 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,\...
18
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0answers
440 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)...
3
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0answers
129 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 ...
3
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1answer
78 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 ...
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0answers
111 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 ...
10
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1answer
343 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 $\Sigma$...
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0answers
36 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 ...
15
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1answer
371 views

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
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0answers
158 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 ...
4
votes
1answer
307 views

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 ...
6
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0answers
186 views

Infinite-dimensional manifold $ΩG$ of loops and the complex projective $n$-space

In a review seminar today, I heard the speaker takes the below for granted, but I have no enough background "Yang-Mills (YM) instantons in 4D can be naturally identified with (i.e. have the same ...
10
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0answers
447 views

How to define (and compute) the Cartan-Killing form of the group of volume-preserving diffeomorphisms?

This question was raised a while ago in a blog post by Terry Tao on the Euler-Arnold equation and he called it "quite tricky". Has anyone in the meantime tried to formulate this question precisely, ...
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0answers
78 views

“Barrier functions” in function spaces [closed]

In general the idea of a "barrier function" (as in the context of "interior point methods") can possibly be thought of as defining a function corresponding to an open subset of $\mathbb{R}^n$ such ...
3
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1answer
89 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
1answer
296 views

Cohomology of Infinite Dimensional Lie Algebra

I am looking for a explicit relation for adjoint cohomology of infinite dimensional Lie algebras (specifically smooth vector fields on a manifold), is it possible to apply Hochschild-Serre spectral ...
6
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0answers
261 views

K-theory of the infinite dimensional projective space

What is the $K$-theory of the category of coherent sheaves on the infinite (countable) dimensional projective space over a field? As far as I know, $K$-theory is oriented; hence this theory should be "...
5
votes
1answer
250 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 $s$ ...
16
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1answer
769 views

Does the image of the exponential map generate the group?

Let $G$ be a connected Fréchet-Lie group and let $\mathfrak g$ be its Lie algebra. Does the image $\exp(\mathfrak g) \subset G$ of the exponential map generate $G$?
10
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0answers
494 views

Differential Forms in Infinite Dimensions

In Kriegl/Michor's book "The convenient setting of global analysis", they define the space of differential $k$-forms on a possibly infinite-dimensional manifold $M$ as the space of smooth sections of ...
19
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2answers
1k views

Infinite dimensional symplectic geometry

Could anyone comment on possible references concerning infinite dimensionsal symplectic manifolds?. I am mainly concerned with hilbert spaces, so i am not interested in the convenient analysis ...
7
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1answer
414 views

Tangent space of the space of smooth sections of a bundle

Let $E\to M$ be a real vector bundle of finite rank over a closed differentiable manifold $M$. Let $C^{\infty}(E)$ denote the space of smooth sections of $E$ and let $e\in C^{\infty}(E)$ be a section. ...
4
votes
0answers
95 views

A criterion for a differential equation to be realized as an Euler-Lagrange equation on the infinite dimensional space

I study PDEs that arise in fluid dynamics in an infinite dimensional Riemannian geometric perspective. For example, Ebin-Marsden(1970) showed that the group of volume preserving diffeomorphisms has an ...
5
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2answers
335 views

Manifold of mappings between $M$ and $N$, with non-compact source $M$

EDIT: Let $M$ and $N$ are two smooth manifold and suppose $N$ is compact but $M$ is not necessarily compact. For my purpose, I just need to consider the case $M=\mathbb R \times S^1$ or $\mathbb R \...
3
votes
2answers
628 views

Integrability conditions for differential equations on $J^\infty$

Is there any result on the existence of solutions of differential equations of the form $$ D_\alpha\Phi([u])=U_\alpha([u])\Phi([u]), $$ where $[u]$ is an element of an infinite dimensional bundle $J^\...
5
votes
1answer
121 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 ...
1
vote
1answer
285 views

Is stable manifold contained in the global attractor?

I am reading James C. Robinson's book "Infinite-dimensional dynamical system -- an introduction to dissipative parabolic PDEs and the theory of global attractors". I have a question regarding the ...
3
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0answers
216 views

Is there a notion of p-forms for $p=\infty$?

On infinite-dimensional manifolds, is there a sensible notion of p-forms with infinite p? Of volume forms? Is there a version of Stokes' integration formula?
2
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1answer
292 views

the tangent space $T_J\mathcal J^k$ of the space of $\omega$-compatible almost complex structure

Let $(M,\omega)$ is a symplectic manifold, and you can assume it is compact if necessary. Denote by $\mathcal J^k = \mathcal J^k(M, \omega)$ the set of all $\omega$-compatible almost complex ...
3
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0answers
159 views

infinite-dimensional transversality theorem and its application on the universal moduli space of pseudo-holomorphic curves

We would like to discuss Proposition 3.2.1 in McDuff&Salamon's book "J-holomorphic Curves and Symplectic Topology(Second Endition)". Let me first remind you some background. Let $\Sigma$ be a ...
2
votes
3answers
366 views

why is $W^{k,p}(\Sigma, u^*TM)$ the tangent space of $W^{k,p}(\Sigma,M)$ at $u$

Let $\Sigma$ and $M$ be two smooth manifolds with $\Sigma$ compact. It is a well-known fact that $W^{k,p}(\Sigma, u^*TM)$ is the tangent space of $W^{k,p}(\Sigma,M)$ at $u$. (You can replace these ...
2
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0answers
316 views

Is the infimum of the p-norm distance between orthogonal vectors in the unit sphere of $\ell^n_p$ equal to the Schaffer constant of $\ell^n_p$?

For $1\leq p<\infty$, we denote by $\ell_p^n$ the vector space $\mathbb{R}^n$ endowed with the p-norm $$\|(a_1,\dots,a_n)\|_p=\left(\sum_{i=1}^n|a_i|^p\right)^{\frac{1}{p}}.$$ For a normed space $...
2
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0answers
98 views

Complex Lie inverse Galois problem

My question is about the inverse Galois problem for infinite dimensional complex manifolds. If $K$ is the field of meromorphic functions over a complex manifold $M$ and $G$ is a finite or infinite ...
1
vote
1answer
881 views

About covariance operators for probability distributions on a function space

Feel free to restrict the function space to a Hilbert space or to a RKHS. Given a probability distribution on it when can we define a ``covariance operator" for it and when would it also have a well-...