All Questions
11 questions
8
votes
1
answer
230
views
The closure of the space of Riemannian metrics with a fixed isometry class
Let $M$ be a closed manifold, and let $\mathscr{M}$ be the space of all Riemannian metrics over $M$. It is known that this is a Fréchet manifold. Consider also $\mathscr{D}$ the diffeomorphisms group ...
- 808
3
votes
0
answers
109
views
"Practical" references on mapping spaces as infinite-dimensional manifolds
I am studying spaces of the form $C^{k}(\mathcal{M},\mathcal{N})$ between manifolds ($k=\infty$ allowed) and I am looking for extensive references, especially analysing their topology and smooth ...
- 1,171
11
votes
5
answers
2k
views
Ricci Curvature in infinite dimensions?
Is there a good notion of "Ricci curvature" in infinite dimensions?
My intuitive understanding of Ricci curvature is that it is some kind of an "average" of the curvature tensor over "different ...
- 63.9k
6
votes
1
answer
199
views
A smooth map on a Banach manifold whose pointwise rank is finite but its rank is not globally bounded
Is there a connected Banach manifold $M$ and a smooth map $f:M \to M$ such that the rank of $Df_x$ is finite for every $x\in M$ but this rank is not uniformly bounded
- 3,669
10
votes
1
answer
377
views
Smooth vector fields on a surface modulo diffeomorphisms
Let $\Sigma$ be a two-dimensional connected smooth manifold without boundary. (Feel free to assume it is compact and orientable.)
Let $\mathcal{X}(\Sigma)$ denote the smooth vector fields on $\Sigma$...
- 28.9k
12
votes
2
answers
2k
views
What is the infinite-dimensional-manifold structure on the space of smooth paths mod thin homotopy?
This question is motivated by the recent paper An invitation to higher gauge theory by Baez and Huerta, and the 2007 paper Parallel Transport and Functors by Schreiber and Waldorf.
Let $M$ be a ...
- 35.5k
7
votes
1
answer
664
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 ...
- 5,824
3
votes
1
answer
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(...
- 6,210
3
votes
1
answer
1k
views
Is it possible to define a structure of differentiable manifold on the Hilbert cube $[0,1]^\mathbb{N}$?
Is it possible to define a structure of differentiable (smooth) manifold on the Hilbert cube $[0,1]^\mathbb{N}$ ?
Has it been done in the literature?
In textbooks, only the Banach case is treated, ...
- 60.5k
1
vote
2
answers
675
views
$\infty$-forms and $\infty$-plectic geometry
Can you have $\infty$-forms on infinite-dimensional manifolds or elsewhere and what are they used for?
- 6,210
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 ...
- 3,584