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9 votes

Group of diffeomorphisms and its tangent space i.e. its Lie algebra

For compact manifolds the identification is a consequence of the exponential law for smooth functions. It states that $$f\colon L\rightarrow C^\infty (K,N)$$ is smooth ($K$ a compact smooth manifold) …
Alexander Schmeding's user avatar
2 votes
Accepted

Reconciling some result about the exponential map, the Chow-Rashevskii theorem, and $\mathrm...

This was a bit long for a comment, thus I post it as an answer. First of all, you have to be very careful with what you actually mean by the exponential here. The flow map to time 1 does NOT exist in …
Alexander Schmeding's user avatar
2 votes
Accepted

$c^\infty$ topology on $L(E, F)$

Since this was a bit long for a comment, I am posting it here as an answer (though it unfortunately does not answer your question per se). It is not clear to me in general how the topologies are rela …
Alexander Schmeding's user avatar
3 votes

Manifolds of maps

To go a bit more into the details (elaborating on Thomas comment above): The result on the differentiability order is classical (going back to Ebin's work in the 60s, culminating in the seminal paper …
Alexander Schmeding's user avatar
18 votes
Accepted

Is the subgroup $\mathrm{Diff}(M,S)$ of $\mathrm{Diff}(M)$ a Lie subgroup?

Before I come to the core of your question concerning the subgroup $\mathrm{Diff}(M,S)$ let us first clarify the terms Lie group and subgroup: In general there are many different concepts, especially …
Alexander Schmeding's user avatar
3 votes
Accepted

Relation between locally convex calculus and Kriegl & Michor's "convenient setting"

It is actually well known that every smooth map in the Michal-Bastiani (MB) sense (what you call "differential calculus on locally convex spaces") is also smooth in the convenient sense. For a quick …
Alexander Schmeding's user avatar