# Questions tagged [infinite-dimensional-manifolds]

The infinite-dimensional-manifolds tag has no usage guidance.

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### 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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**0**answers

303 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

74 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

84 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 ...

**0**

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**0**answers

64 views

### Adjoint Representation of Witt algebra

The form of Witt algebra (algebra of smooth vector fields on circle) is:
$[L_{m},L_{n}]=(m-n)L_{m+n};\,\,\ m, n\in \mathbb{Z}$,
this Lie algebra is simple or semi-simple? Also, I am studying adjoint ...

**1**

vote

**1**answer

209 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

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193 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 "...

**2**

votes

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76 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 $s$ ...

**13**

votes

**1**answer

478 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**

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210 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**

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**2**answers

845 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**

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**1**answer

229 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

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77 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

305 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

279 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^\...

**3**

votes

**0**answers

69 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 ...

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vote

**1**answer

172 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

211 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

**1**answer

187 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

**0**answers

105 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 ...

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votes

**3**answers

225 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

198 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

93 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 ...

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vote

**1**answer

391 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

162 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 $M$ ...

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187 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

488 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. ...

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votes

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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\...

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votes

**2**answers

326 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

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296 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}(\...

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votes

**1**answer

221 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

**1**answer

286 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 ...

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**1**answer

316 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

352 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-...

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**1**answer

220 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

227 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

227 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
$$
\...

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votes

**2**answers

297 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

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

**1**answer

377 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 ...

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**3**answers

530 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 \...

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**0**answers

213 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

196 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

252 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 ...

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**0**answers

49 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 ...

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**2**answers

781 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 ...

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**1**answer

392 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**

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**0**answers

574 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 $...

**-3**

votes

**2**answers

738 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

333 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((...