Questions tagged [infinite-dimensional-manifolds]
The infinite-dimensional-manifolds tag has no usage guidance.
102
questions
15
votes
0answers
220 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
votes
0answers
101 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
votes
1answer
60 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 ...
1
vote
0answers
82 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
votes
1answer
317 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 $\Sigma$...
1
vote
0answers
32 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
votes
1answer
322 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
votes
0answers
145 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
162 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
votes
0answers
175 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
votes
0answers
331 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, ...
1
vote
0answers
75 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
votes
1answer
86 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
238 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
votes
0answers
215 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 "...
3
votes
1answer
132 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 $s$ ...
15
votes
1answer
573 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 $G$?
9
votes
0answers
297 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 ...
18
votes
2answers
948 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 ...
6
votes
1answer
282 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. ...
3
votes
0answers
83 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
votes
2answers
317 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
383 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^\...
4
votes
1answer
96 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
203 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
votes
0answers
213 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
votes
1answer
208 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
votes
0answers
127 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
256 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
votes
0answers
244 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
votes
0answers
96 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
580 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-...
4
votes
0answers
169 views
Singular symplectic reduction in infinite dimension
In 1991, Sjamaar and Lerman [1] introduced the notion of stratified symplectic spaces. Namely, if $M$ is a symplectic manifold and $G$ a Lie group acting properly (but not necessarily freely) on $M$ ...
8
votes
0answers
188 views
Holomorphic contractibility of GL(H)?
Kuiper's theorem is well-known to give the triviality of the homotopy groups of ${\rm GL}(\mathcal{H})$ for $\mathcal{H}$ a (separable) infinite-dimensional complex Hilbert space. Work of Palais later ...
14
votes
1answer
634 views
Are infinite simplicial complexes all manifolds?
Are infinite dimensional simplicial complexes manifolds locally modeled on $\mathbb R^\infty=\operatorname{colim}\mathbb R^n$? If they are homotopy equivalent, are they homeomorphic?
Of course not. ...
40
votes
6answers
3k views
Does $\mathbb C\mathbb P^\infty$ have a group structure?
Does $\mathbb C\mathbb P^\infty$ have a (commutative) group structure? More specifically, is it homeomorphic to $FS^2$, (the connected component of) the free commutative group on $S^2$?
$\mathbb C\...
7
votes
2answers
335 views
Symplectic Reduction on infinite dimensional manifolds
Let $X$ be a compact, oriented Riemann manifold. Let $\pi_{P}: P \rightarrow X$ be a principal $G$-bundle over $X$, for a compact Lie group $G$. Let $(M, \omega)$ be a symplectic manifold endowed with ...
6
votes
2answers
336 views
Relationship between the Lie functor applied to a Lie group action, and the fundamental vector field mapping?
Let $M$ be a smooth manifold, and $G$ a Lie group with Lie algebra $\mathfrak{g}$. The Lie algebra of the diffeomorphism group of $M$ is the Lie algebra of vector fields on $M$; that is $\text{Lie}(\...
4
votes
1answer
242 views
Some notational questions regarding tangent vectors
I am not a specialist in differential geometry, so I have some difficulties in finding the right words for the following natural things:
First of all it seems that there is a lot of nonequivalent ...
6
votes
1answer
333 views
Submersion theorem for smooth tame Frechet manifolds
If $M$ and $N$ are Banach manifolds, $f:M\rightarrow N$ is a smooth map, and $q\in N$ is a regular value, so $f$ is a submersion on $f^{-1}(q)$, it is well known that the level set $f^{-1}(q)$ is a ...
11
votes
1answer
333 views
How many Fréchet manifolds are there?
Clearly the title needs clarifying. Allow me to let "how many" to mean a set larger than a skeleton of the category of Fréchet manifolds and smooth maps, if this category is indeed essentially small.
...
5
votes
1answer
374 views
Kernel of flux homomorphism (Calabi invariant) for volume-preserving maps on a compact manifold
Good morning everybody, I am currently reading through the book of Banyaga "Structure of classical diffeomorphism groups" link, and I am particularly interested in the question of factorizing volume-...
8
votes
1answer
229 views
Closed geodesics in free smooth loop space?
I know very little about these subjects, so I apologise if this is a naive line of inquiry:
Let $M$ be a smooth $n$-dimensional Riemannian manifold. I understand that it is possible to construct an ...
1
vote
1answer
237 views
Relation between locally convex calculus and Kriegl & Michor's “convenient setting”
I have a very general question regarding the book "Kriegl, Michor: The Convenient Setting of Global Analysis.":
Is the differential calculus of locally convex spaces (see here, for instance) ...
4
votes
1answer
239 views
Infinite dimensional Cauchy-Lipschitz theorem [duplicate]
From the answers to Mathoverflow Question "Continuity in Banach space for non-linear maps", it is possible to infer that the assumption of the Cauchy-Lipschitz theorem for the autonomous equation
$$
\...
6
votes
2answers
340 views
Is the strong Whitney topology connected?
$\def\bbR{\mathbb R}\def\ssp{\kern.4mm}$Put more precisely, let $C$ be the set of all continuous functions $f:\bbR\to\bbR$ when
$\bbR$ has its standard order topology. Let $\mathscr T$ be the set of ...
1
vote
1answer
134 views
Restriction of derivations on $C^\infty(X)$
In 'Kriegl, Michor - A convenient setting for global infintite-dimensional analysis', they say that for an element $x$ in a convenient (i.e. Mackey-complete locally convex) space $X$, a bounded ...
2
votes
1answer
441 views
Does Schur's Lemma hold in this case? Regular representations of $S_n$ over $\mathbb R$
Warning I am a physicist and I am not familiar with a lot of the machinary of representation theory.
I consider the regular representation of $\mathbb S_n$ over reals $\mathbb R$ ($\mathbb R \mathbb ...
3
votes
3answers
534 views
What should be considered a finite size of an infinite dimensional space? [closed]
I've got a map between two infinite dimensional spaces, $f: A\to B$, where $A$ seems "larger" than $B$. For the sake of conversation let's assume that $A$ is the set of smooth maps $\mathbb R^3\to \...
2
votes
0answers
221 views
infinite dimensional germs of schemes and tangent spaces
(The question of the type "how to define?")
Let $(R,\mathfrak{m})$ be a local ring over a field $k$ of zero characteristic. Consider the matrices over this ring, $Mat(m,R)$. I think of $Mat(m,R)$ as ...
