Questions tagged [frechet-manifold]

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If a Banach / Fréchet manifold $M$ happens to be a topological vector space, is $M$ just a Banach / Fréchet space?

In finite dimensions, if $M$ is a smooth manifold that happens to be a vector space, then it is indeed just the Euclidean space. I wonder if the same result holds valid in infinite dimensions. More ...
Isaac's user avatar
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13 votes
2 answers
698 views

Smooth Urysohn's lemma on Fréchet spaces

Let $V$ be a Fréchet topological vector space. Let $K_0$ and $K_1$ be two closed subsets which are disjoint. I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$ whose restriction ...
André Henriques's user avatar
1 vote
0 answers
64 views

Banach tori: classification up to Fréchet homeomorphisms

Consider the set $T$ in $l_p$ defined as closure of \begin{equation} T = \{ (x_1,\dotsc,x_n,\dotsc): x_j = \frac{1}{2^{(j/p)}} e^{it_j}, t_j \in \mathbb{R}/\mathbb{Z} \}. \end{equation} This seems to ...
0x11111's user avatar
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7 votes
0 answers
140 views

Are these two concepts of a differential form on the loop space equivalent?

Notation: Let $X$ denote a smooth manifold (without boundary) and define $LX = C^{\infty}(S^1, X)$ to be the loop space on $X$. In the context of loop space homology and the supersymmetric path ...
ChenIteratedIntegral's user avatar
6 votes
0 answers
149 views

Is the space of analytic sections of a vector bundle a Fréchet space?

Let $\pi : E \to M$ a smooth vector bundle of finite rank, where both $E$ and $M$ are finite dimensional smooth manifolds, and $M$ is compact. I already know that the space $\Gamma^\infty(M,E)$ of ...
Eduardo Longa's user avatar
1 vote
0 answers
55 views

When does an analytic submanifold descend to the quotient?

Let $\pi : M \to B$ be a smooth principal bundle with group $G$, where $M$ is an analytic (Fréchet, in my case) manifold, $B$ is a smooth (Fréchet) manifold and $G$ is a smooth (Fréchet) Lie group. ...
Eduardo Longa's user avatar
2 votes
0 answers
234 views

Why is the space of smooth sections complete?

This page is about the space of sections: Let $E \stackrel{\mathrm{fb}}{\rightarrow} \Sigma$ be a smooth vector bundle. On its real vector space $\Gamma_{\Sigma}(E)$ of smooth sections consider the ...
amilton moreira's user avatar
2 votes
1 answer
109 views

Smooth dependence in the fixed point theorem between complete Fréchet manifolds

Let $X,Y$ be complete metric spaces, and let $\Sigma:X\times Y\rightarrow Y$ be a continous mapping which satisfies the following property: there exists a $C<1$, such that for all $x\in X$ and $y_{...
MySheperd's user avatar
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1 vote
0 answers
69 views

What is the relationship between a metric and the Frechet structure

Let $M$ be a smooth, closed m dimensional manifold. Let $K\subset M$ be a smooth stratifold inside $M$. I'd like to show that $\mathop{Diff}(M,K)$ is a Frechet submanifold of $\mathop{Diff}(M)$. ...
Daniel H. Hartman's user avatar
4 votes
0 answers
174 views

Global analysis on punctured surfaces

Global analysis on open manifolds seems pretty hard. For one, the space of $C^{n,\alpha}$ functions on an open manifold need not be a tame Fréchet space (see the post Are smooth functions tame? for ...
user158773's user avatar
1 vote
0 answers
60 views

Is there a version of Hamilton's infinite dimensional family of implicit function theorems which gives us a submersion map?

Hamilton, in his notes on "Inverse Function Theorem of Nash and Moser" states a theorem(1.1.3 on Page 172), where a given nonlinear map between tame Frechet spaces is locally surjective, if ...
Harshavardhan's user avatar
4 votes
3 answers
545 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 \...
vmist's user avatar
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7 votes
0 answers
129 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 ...
Martin Pinsonnault's user avatar
2 votes
1 answer
710 views

Metrizability of topology of compact convergence

Let $X$ be a separable metric space and $Y$ be a second-countable $\sigma$-compact Hausdorff space. Then the compact-convergence (compact-open) topology on $C(Y,X)$ is metrizable with metric $$ d(f,g)...
ABIM's user avatar
  • 4,989
3 votes
1 answer
123 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 ...
cr1t1cal's user avatar
  • 755
5 votes
1 answer
482 views

What is the connection between Frechet Lie groups and Lie algebras?

An ordinary Lie group has a differentiable manifold structure, i.e. it is locally isomorphic to a finite-dimensional Euclidean space. A Frechet Lie group, on the other hand, has a Frechet manifold ...
Keshav Srinivasan's user avatar
1 vote
0 answers
41 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 ...
Harshavardhan's user avatar
1 vote
0 answers
57 views

Tangent space and a subset of a tame Lie group

I am curious if the set of all orientation-preserving diffeomorphisms with a given rotation number is a tame Lie subgroup or a tame submanifold of all orientation-preserving diffeomorphisms on the ...
Hua Ying's user avatar
8 votes
1 answer
765 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
  • 3,064
7 votes
1 answer
601 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
  • 366
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
  • 33.4k
2 votes
1 answer
263 views

Smooth curves in a Frechet space

Is the space $C^{\infty}([0,1];C^{\infty}(S^1))$ equal with the space $C^{\infty}([0,1]\times S^1)$ ? I am interested in characterizing the smooth curves in the space $C^{\infty}(S^1)$ where $S^1$ is ...
Emanuel's user avatar
  • 21
7 votes
2 answers
597 views

Inverse of partial differential operator as a smooth tame map

Tameness for maps is one of the main ingredients for the Nash-Moser inverse function theorem. A linear map $f: X \to Y$ between Fŕechet spaces with fixed seminorms is called tame if we have an ...
Tobias Diez's user avatar
  • 5,442
4 votes
0 answers
101 views

Convex subsets of infinite dimensional spaces up homeomorphism

Let $C$ be a convex, infinite-dimensional, non-locally-compact subset of a separable Frechet space. If $C$ is a closed subset (or more generally, if $C$ is completely metrizable), then it is known ...
Igor Belegradek's user avatar
2 votes
2 answers
302 views

evaluation map $ev_t$ on loop space

Considering parameter of $S^1$ as $t$, we define. $$ev_t: C^\infty(S^1, \mathbb R^n)\to \mathbb R^n$$ $$ev_t(\gamma):=\gamma(t)$$ I am looking for a possible topology on $C^\infty(S^1,\mathbb R^n)$ ...
Jonujohn's user avatar
  • 217
0 votes
1 answer
152 views

Constant symplectic structure

Let $E$ be a Frechet space and $\mathcal{F}$ be a non-degenerate bounded skew symmetric bilinear map $\mathcal{F}: E\times E\to \mathbb R$ on $E$. We can identify $TE$ with $E\times E$, with this ...
Jonujohn's user avatar
  • 217
3 votes
2 answers
198 views

Space of differential operators

Let $A$, $B$ be two smooth vector bundles of finite rank over a smooth manifold $M$. Let $Diff(A,B)$ be the space of differential operators from $A$ to $B$. Can I talk about "the space of smooth maps ...
tujunwu's user avatar
  • 85
14 votes
2 answers
1k views

Are smooth functions tame?

I know the article of Hamilton on the inverse function theorem of Nash and Moser (with the same title) where he proves that $C^\infty(M)$ is a tame Fréchet space, when $M$ is closed or compact with ...
Matthias Ludewig's user avatar
30 votes
7 answers
4k views

Intuition for failure of Implicit Function theorem on Frechet Manifolds

When dealing with moduli spaces of, say connections or metrics, I am using the notions of Frechet spaces/manifolds/groups. I have become familiar with Banach manifolds (I think), but Frechet manifolds ...
Chris Gerig's user avatar
  • 17.1k
35 votes
4 answers
5k views

How are infinite-dimensional manifolds most commonly treated?

I originally posted this question on StackExchange, where it was suggested I post here. It was also suggested I read about Hilbert manifolds and Fréchet manifolds. Nevertheless, I am still looking for ...
Daan Michiels's user avatar
4 votes
2 answers
288 views

On the definition of ‘smooth vectors’ in Rieffel's “Deformation Quantization for Actions of $ \mathbb{R}^{d} $”.

On the first page of Chapter 1 of Rieffel's Deformation Quantization for Actions of $ \mathbb{R}^{d} $, Rieffel defines a family of seminorms on the space $ A^{\infty} $ of smooth vectors of a Fréchet ...
Leonard's user avatar
  • 816
2 votes
0 answers
309 views

What are the current possibilities for infinite-dimensional manifolds? [closed]

According to wikipedia, by a theorem of Henderson '69, infinite-dimensional Frechet Manifolds embed as open subspaces of Hilbert Space. They need to be seperable & metric. They are generalisations ...
Mozibur Ullah's user avatar
4 votes
1 answer
414 views

Smooth functions tangent to the leaves of a foliation

Given two smooth manifolds $M$ and $N$, it is known that if $M$ is compact, then $C^\infty(M,N)$ is a Fréchet manifold whose tangent space at $f \in C^\infty(M,N)$ is the space $$T_f C^\infty(M,N) = \...
HYL's user avatar
  • 1,809
8 votes
3 answers
1k views

Loop space: De Rham cohomology

How to calculate the DeRham cohomology of the free loop space $LM= C^\infty(S^1,M)$ as a Frechet manifold?. Edit: It will be enough for me to know: When $H^1_{DR}(LM)$ is not $\{0\}$. Bounty ...
Jonujohn's user avatar
  • 217
5 votes
1 answer
513 views

A fact about finite-dimensional manifolds I fear does not hold for Frechet manifolds (what's new?)

Let $M$ be a manifold equipped with a pair of surjective submersions $N_1 \stackrel{p_1}{\leftarrow} M \stackrel{p_2}{\rightarrow} N_2$ where $dim N_1 = dim N_2 = n$. Then we can find, for any point $...
David Roberts's user avatar
  • 33.4k
3 votes
0 answers
562 views

Loop space: various manifold structure.

While reading articles, Sometimes i see collection of all smooth loops as hilbert manifold(pre). Sometime i see this space as banach manifold. Sometime i see this sapce as nuclear frechet space. Can ...
Jonujohn's user avatar
  • 217
6 votes
0 answers
349 views

Regular maps between Fréchet manifolds and pullbacks

An oft-used example of a regular map of finite-dimensional smooth manifolds is a submersion. We have the well-known result that the pullback of a submersion exists and is a submersion. For Fréchet ...
David Roberts's user avatar
  • 33.4k
6 votes
3 answers
729 views

Internal equivalence implies weak equivalence for Frechet Lie groupoids?

It is a known theorem that an internal equivalence of Lie groupoids (finite dimensional manifolds!) - that is an equivalence in the 2-category of Lie groupoids, smooth functors and transformations - ...
David Roberts's user avatar
  • 33.4k
11 votes
3 answers
1k views

Induced map on path manifolds: is it a submersion?

Consider the following claim: Let $p:M \to N$ be a (surjective) submersion of finite-dimensional smooth manifolds. Let $J$ denote one of $[0,1],\ [0,1),\ (0,1]$. Then $p_*:M^J \to N^J$ is a ...
David Roberts's user avatar
  • 33.4k
3 votes
1 answer
436 views

Ind-Frechet manifolds?

Short version: has anyone done geometry on something that is the formal filtered colimit of Frechet manifolds? Longer version: A colleague and I came up with a concept today that seems like we ...
David Roberts's user avatar
  • 33.4k
8 votes
2 answers
835 views

frechet manifolds book

hi, does anyone know a good book or some lecture notes on the theory of frechet manifolds ?
2 votes
1 answer
228 views

Complement of a closed star-shaped subset in a Frechet space

Let $U$ be the complement of a closed star-shaped subset in a separable infinite-dimensional Frechet space. Since every separable Frechet space is homeomorphic to $l_2$, one knows that $U$ is a ...
Igor Belegradek's user avatar
6 votes
3 answers
774 views

Two notions of tangent vector for a Fréchet manifold

Let $X$ be a Frechet or Banach manifold. We can define tangent vectors by equivalence classes of smooth curves. But, we could also define them as derivations of germs of smooth functions. Do these two ...
David Carchedi's user avatar