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Questions tagged [banach-manifold]

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4
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1answer
145 views

null-bordant vs null-homologous sub-manifolds of $\infty$-d spaces/CW complexes

$\require{AMScd}$ Preliminaries: Let $\Sigma$ be a closed manifold, $X$ be a CW complex and $f:\Sigma \to X$ be a map. We say that the pair $(\Sigma,f)$ is null-homologous (over $\mathbb{Z}_2$) if $...
4
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0answers
166 views

The structure of Banach manifolds in symplectic geometry

Let $M$ be a symplectic manifold, and let $L_0$ and $L_1$ be Lagrangian submanifolds which transverse to each other. In Floer theory, we need to consider a Banach manifold $\mathcal B$ of maps $u:\...
5
votes
1answer
394 views

What would be the cotangent bundle of a Banach manifold?

Reference: Lang - Differential manifolds p.123 Quick question: Lang defines the cotangent bundle as the dual vector bundle of the tangent bundle, but shouldn't there be additionally a somewhat ...
2
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1answer
192 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 ...
2
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3answers
228 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
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0answers
71 views

Second Countability hypothesis for a Banach manifold

Is the second countability hypothesis necessary to rigoruosly define a Banach Manifold (say in infinite dimension)? In the finite-dimensional theory of manifolds, that request is included in the ...
4
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0answers
274 views

Intuition: Smooth functions on Banach Spaces

On finite-dimensional vector spaces, we all have a reasonable idea of which functions are likely to be $C^1$ or smooth. When it comes to differentiation on Banach spaces, I find that my `intuition' ...
1
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0answers
96 views

Projective tensor product continuous?

For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous, where $B(V)$ denotes the set of all linear bounded endomorphisms with operator ...
2
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0answers
259 views

Structure of $C^k$ ($k<\infty$)Riemannian metrics on a manifold

$M$ is a smooth manifold. It's known that if $M$ is compact, then the space of smooth Riemannian metrics has a Frechet manifold structure. For the space of $C^k$($k<\infty$) Riemannian metrics, ...
2
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2answers
277 views

Geodesic on Banach Manifold [closed]

Is there a way of defining a geodesic on a Banach Manifold $M$ which is not itself a Hilbert Manifold?
8
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2answers
241 views

Banach manifold of paths with endpoints on submanifolds

Fix a Riemannian metric on a manifold $M$. Suppose that we fix two points $x,y \in M$. We start with the space $C^{\infty}_{\searrow}(x,y) = \left\{\gamma: \mathbb{R}\to M:\,\lim_{t\to-\infty}\...
0
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1answer
79 views

Embedding Riemmanian Manifold Linearly

Given a Hilbert Manifold $M$ does there exist a smooth map into some very large Hilbert space taking geodesics to straight lines?
6
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0answers
270 views

Reference for the Banach Manifold structure of $C^k(M,N)$

I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following: Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set $C^...
4
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0answers
190 views

Sobolev spaces of maps between manifolds and the Palais-Smale Condition

I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts: Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean space....
3
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1answer
1k views

Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space?

Suppose that $X$ is a compact, finite dimensional manifold and $Y$ is an infinite dimensional, second countable ($C^\infty$-)Banach manifold. Let $\nu \in \mathbb{N}$. Question: Is the space $C^\nu(...
2
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1answer
216 views

Manifold structure for the set of solutions to a first order elliptic system?

Consider a bounded domain $S\subset R^2$ and an elliptic system of two first order PDEs, namely a generalization of the Cauchy-Riemann system allowing nonconstant coefficients and lower order terms. ...
27
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4answers
2k 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 ...
5
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1answer
400 views

Banach Manifold

Let $M$ and $N$ be closed manifolds. Is it true that $C^{k}(N,M)$, which is the space of functions $f: N\to M$ such that $f\in C^{k}$, is a $C^{\infty}$ Banach manifold? If so, can you help me to ...
20
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1answer
2k views

Does this Banach manifold admit a Riemannian metric?

First, the question; after, the motivation. Consider 27.6 (pdf pp. 262-263) in The convenient setting of global analysis (AMS, 1997), and, in particular, the example given at the end of it, which ...
22
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1answer
1k views

Non-regular Connected Hausdorff Banach Manifold

After reading this MO post, I am wondering: Is every (connected) Hausdorff Banach manifold a regular space? Though unjustified, page 53 of this paper nonchalantly states: "Note that a Hausdorff ...
21
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2answers
2k views

Does it make sense to talk about smooth bundles of Hilbert spaces?

Is there a notion of "smooth bundle of Hilbert spaces" (the base is a smooth finite dimensional manifold, and the fibers are Hilbert spaces) such that: 1• A smooth bundle of Hilbert spaces ...
16
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2answers
3k views

Are Banach Manifolds intrinsically interesting?

In the introduction to 'A convenient setting for Global Analysis', Michor & Kriegl make this claim: "The study of Banach manifolds per se is not very interesting, since they turnout to be open ...
1
vote
1answer
821 views

Is there an Error on pg. 17 of Tromba's “Teichmuller Theory in Riemannian Geometry”?

I'm pretty sure that this is a minor error, but I could use some help here. So the book I'm referring to in the title is this book (MR1164870). On pg. 16-17, he is proving that the space of almost ...
6
votes
3answers
494 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 ...