# Questions tagged [frechet-manifold]

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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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)... 3 votes 1 answer 117 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 ... 5 votes 1 answer 429 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 ... 1 vote 0 answers 40 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 ... 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 ... 8 votes 1 answer 686 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. ... 7 votes 1 answer 562 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 513 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. ... 2 votes 1 answer 255 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 ... 7 votes 2 answers 553 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 ... 4 votes 0 answers 98 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 ... 2 votes 2 answers 292 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^nev_t(\gamma):=\gamma(t)$$I am looking for a possible topology on C^\infty(S^1,\mathbb R^n) ... 0 votes 1 answer 147 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 ... 3 votes 2 answers 195 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 ... 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 ... 29 votes 7 answers 3k 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 ... 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 ... 4 votes 2 answers 283 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 ... 2 votes 0 answers 304 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 ... 4 votes 1 answer 409 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) = \...
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 ...