Questions tagged [infinite-dimensional-manifolds]

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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, ...
S.Surace's user avatar
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"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 ...
gradstudent's user avatar
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3 votes
1 answer
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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 ...
Netivolu's user avatar
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1 answer
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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 ...
Hamidreza Safari's user avatar
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 "...
Mikhail Bondarko's user avatar
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 $s$ ...
Jacob Lu's user avatar
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16 votes
1 answer
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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 $G$?
Jarek Kędra's user avatar
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10 votes
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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 ...
Matthias Ludewig's user avatar
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 ...
Javier's user avatar
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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. ...
Bilateral's user avatar
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5 votes
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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 ...
Jae Min Lee's user avatar
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 \...
Hang's user avatar
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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^\...
Javier's user avatar
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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 ...
Matthias Ludewig's user avatar
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 ...
kevin's user avatar
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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?
Arnold Neumaier's user avatar
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 ...
Hang's user avatar
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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 ...
Hang's user avatar
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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 ...
Hang's user avatar
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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 $...
André Porto's user avatar
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 ...
Antoine Balan's user avatar
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-...
gradstudent's user avatar
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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 $M$ ...
SHP's user avatar
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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 ...
David Roberts's user avatar
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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. ...
Ben Wieland's user avatar
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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\...
Ben Wieland's user avatar
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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 ...
Varun's user avatar
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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}(\...
ಠ_ಠ's user avatar
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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 ...
erz's user avatar
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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 ...
mdg's user avatar
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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. ...
David Roberts's user avatar
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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-...
guido giuliani's user avatar
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 ...
Spencer's user avatar
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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) ...
Bipolar Minds's user avatar
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 $$ \...
Bazin's user avatar
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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 ...
TaQ's user avatar
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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 ...
Bipolar Minds's user avatar
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 ...
vishmay's user avatar
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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 \...
Michael's user avatar
  • 2,175
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 ...
Dmitry Kerner's user avatar
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\...
Josh Burby's user avatar
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 ...
David Hornshaw's user avatar
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 ...
Alessandro Gentile's user avatar
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 ...
Wintermute's user avatar
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 ...
Alex M.'s user avatar
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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 $...
Tomasz Lenarcik's user avatar
-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.
user8991's user avatar
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((...
Mostafa's user avatar
  • 403
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, ...
Mostafa's user avatar
  • 403
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-...
Sebastien Palcoux's user avatar