Questions tagged [frechet-manifold]
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42
questions
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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 ...
- 42.2k
1
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0
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64
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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 ...
- 353
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0
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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 ...
5
votes
0
answers
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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 ...
- 2,075
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0
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53
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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. ...
- 2,075
3
votes
0
answers
219
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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 ...
2
votes
1
answer
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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_{...
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0
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69
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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 $\mathop{Diff}(M)$. ...
4
votes
0
answers
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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 ...
- 153
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0
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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 ...
- 183
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votes
3
answers
514
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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 \...
- 847
7
votes
0
answers
126
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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 ...
2
votes
1
answer
612
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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)...
- 5,001
3
votes
1
answer
123
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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 ...
- 755
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votes
1
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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 ...
- 4,489
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0
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40
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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 ...
- 183
1
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0
answers
57
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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 ...
- 37
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votes
1
answer
723
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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. ...
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votes
1
answer
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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
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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.
...
- 33.2k
2
votes
1
answer
258
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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 ...
- 21
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2
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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 ...
- 5,412
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votes
0
answers
101
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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 ...
- 28.2k
2
votes
2
answers
296
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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)$ ...
- 207
0
votes
1
answer
150
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 ...
- 207
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votes
2
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196
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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 ...
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votes
2
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1k
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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 ...
- 9,481
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votes
7
answers
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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 ...
- 16.9k
35
votes
4
answers
5k
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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 ...
- 645
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2
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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 ...
- 816
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0
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306
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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 ...
- 2,023
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1
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411
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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) = \...
- 1,809
8
votes
3
answers
1k
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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 ...
- 207
5
votes
1
answer
509
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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 $...
- 33.2k
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553
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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 ...
- 207
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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 ...
- 33.2k
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votes
3
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728
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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 - ...
- 33.2k
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3
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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 ...
- 33.2k
3
votes
1
answer
427
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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 ...
- 33.2k
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votes
2
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826
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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
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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 ...
- 28.2k
6
votes
3
answers
764
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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 ...
- 15.3k