Questions tagged [infinite-dimensional-manifolds]
The infinite-dimensional-manifolds tag has no usage guidance.
142
questions
11
votes
0
answers
619
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
0
answers
82
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
1
answer
197
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
1
answer
374
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
0
answers
365
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
1
answer
363
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
votes
1
answer
1k
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
votes
0
answers
731
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 ...
21
votes
2
answers
2k
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 ...
8
votes
1
answer
768
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. ...
5
votes
0
answers
127
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
2
answers
366
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
2
answers
721
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
1
answer
177
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
1
answer
414
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
0
answers
220
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?
3
votes
1
answer
378
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 ...
4
votes
0
answers
190
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
3
answers
501
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
0
answers
410
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
0
answers
105
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
1
answer
1k
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
0
answers
202
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
0
answers
219
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
1
answer
1k
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. ...
44
votes
6
answers
4k
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
2
answers
524
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
2
answers
691
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
1
answer
385
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 ...
7
votes
1
answer
604
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
1
answer
519
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
1
answer
550
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
1
answer
281
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
1
answer
319
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
1
answer
363
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
$$
\...
9
votes
2
answers
631
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
1
answer
151
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
1
answer
733
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
3
answers
572
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
0
answers
279
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 ...
4
votes
0
answers
217
views
Casimirs of Poisson brackets obtained via Poisson reduction
Suppose that a Lie group $G$ acts freely and properly on a symplectic manifold $P$ via symplectomorphisms. Suppose further that we have at our disposal an $\text{Ad}^*$-equivariant momentum map $\mu:P\...
2
votes
1
answer
472
views
Condition for infinite dimensional complex manifold to be Kähler by pullback form
For a finite dimensional Kähler manifold $M$, there is the condition that if $N\xrightarrow{f}M$ is a holomorphic map with maximal rank at each point, then $N$ is Kähler with the pullback form induced ...
2
votes
0
answers
52
views
Topology of fibers of operators under C^{\infty} convergence
A smooth family of maps $f_t : L^2 (\mathbb{R}) \rightarrow \mathbb{R}^{n}$ is given, with $t \in (0,1]$. Suppose that when $t \rightarrow 0$ the family restricted to balls of given radius converges ...
10
votes
2
answers
1k
views
Infinite dimensional Riemannian geometry
My current research has brought me into an area the requires me to learn some infinite dimensional Riemannian and Kähler geometry. Can someone recommend some good books or survey articles to help me ...
10
votes
1
answer
450
views
Monograph or rich survey on infinite-dimensional Riemann manifolds
I'm working with the space of smooth curves $\mathcal{C}$ in a smooth manifold $M$, having (different, pre-determined) fixed endpoints. I'd like to endow it with a Riemann structure (I already have a ...
1
vote
0
answers
788
views
Does the coordinate ring of affine variety admit a structure of infinite dimensional variety?
We work in the category of algebraic varieties over
some algebraically closed field $k$.
By infinite dimensional variety I mean a filtration:
$$
V_0\subset V_1\subset V_2\subset\ldots
$$
where each $...
-2
votes
2
answers
1k
views
Are there examples of compact infinite dimensional manifolds? [closed]
Are there known examples of compact infinite dimensional manifolds?
The word "manifold" is important.
4
votes
2
answers
399
views
Can we define smooth diffeomorphisms on the Hilbert cube $[0,1]^\mathbb{N}$ ? Has it been done in the literature?
Can we define smooth diffeomorphisms on the Hilbert cube$[0,1]^\mathbb{N}$ ? Has it been done in the literature?
For example, let $E$ be the set of bijections $f$ of the Hilbert cube, such that $f((...
3
votes
1
answer
1k
views
Is it possible to define a structure of differentiable manifold on the Hilbert cube $[0,1]^\mathbb{N}$?
Is it possible to define a structure of differentiable (smooth) manifold on the Hilbert cube $[0,1]^\mathbb{N}$ ?
Has it been done in the literature?
In textbooks, only the Banach case is treated, ...
5
votes
2
answers
551
views
Homotopy problem for infinite dimensional topological space II
This post here is a specification of this post.
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of intrinsic metric spaces verifying :
$X_{n}$ have topological dimension $n$.
$X_{n+1}$ is n-...