All Questions
9 questions
11
votes
1
answer
560
views
Different smooth structures on the infinite jet bundle (for the purposes of calculus of variations)
Let $\pi:Y\rightarrow X$ be a (smooth, finite dimensional) fibred manifold. Since no other fibrations will be considered on $Y$, I will identify $(Y,\pi,X)$ with $Y$. The finite order jet bundles are ...
- 2,792
2
votes
1
answer
120
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_{...
- 808
2
votes
2
answers
446
views
Reconciling some result about the exponential map, the Chow-Rashevskii theorem, and $\mathrm{Diff}_0(M)$
Let $M$ be a $C^{\infty}$ manifold $C^{\infty}$-diffeomorphic to $\mathbb{R}^d$. I've recently come across some results which I'm trying to reconcile. Let $\mathfrak{X}(M)$ denote the set of ...
- 5,405
5
votes
1
answer
710
views
Smooth structure on the space of sections of a fiber bundle and gauge group
Let $\xi$ be a fiber bundle $F\hookrightarrow E\to B$ (where every space is smooth, T2 and second countable), let $\Gamma(\xi)$ be the space of smooth sections. We can complete $\Gamma(\xi)$ with ...
- 2,533
-2
votes
2
answers
1k
views
Are there examples of compact infinite dimensional manifolds? [closed]
Are there known examples of compact infinite dimensional manifolds?
The word "manifold" is important.
- 73
4
votes
2
answers
406
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((...
- 403
1
vote
1
answer
283
views
Cotangent bundle in the category of locally convex spaces
I'm trying to understand the definition of a differential form on $M$ in the context of Fréchet spaces or, more generally, locally convex spaces. The standard procedure defines a k-form as a map $\...
- 5,824
24
votes
3
answers
2k
views
The third axiom in the definition of (infinite-dimensional) vector bundles: why?
Serge Lang's Differential and Riemannian Manifolds is a no doubt the best available reference for the theory of not-necessarily-finite-dimensional differential manifolds, but unfortunately it suffers ...
- 243
5
votes
1
answer
1k
views
Are smooth functions on an uncountable sum continuous?
Consider the linear space $\sum_{\mathbb{R}} \mathbb{R}$. As in the Frolicher-Kriegl-Michor view, we make this into a Frolicher space as follows.
Equip it with the locally convex topology of the ...
- 26.8k