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consider a smooth manifold and the group of diffeomorphisms (or (local) isometries in case of riemannian manifolds) $\varphi:M \rightarrow M$. How can one define a smooth structure on this group, s.t. it becomes a Lie group?

Regards

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    $\begingroup$ I think in the smooth case you get infinite-dimensional generalizations of Lie groups. $\endgroup$ Commented Jun 17, 2013 at 5:14
  • $\begingroup$ Ok, I don't know much about infinite dimensional manifolds. Do you have a reference where I can read about this kind of manifolds? $\endgroup$
    – Braten
    Commented Jun 17, 2013 at 5:15
  • $\begingroup$ johncarlosbaez.wordpress.com/2012/03/12/… $\endgroup$ Commented Jun 17, 2013 at 5:30
  • $\begingroup$ The diffeomorphism entry in the wikipedia tells you why the group of diffeomorphisms of a compact manifold is a Banach manifold. $\endgroup$ Commented Jun 17, 2013 at 17:03
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    $\begingroup$ Just a comment to the comment: If you are looking at "smooth" diffeomorphisms, then the diffeomorphism group of a compact manifold ist never Banach but only Frechet. $\endgroup$
    – Tom
    Commented Feb 13, 2015 at 13:18

2 Answers 2

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This is shamelessly ripped from "Milnor, Remarks on infinite dimensional Lie Groups. In: Relativity, Groups and Topology II."

Endow $M$ with an arbitrary Riemannian metric (always possible via the wizardy of partitions of unity) and let $V = \operatorname{Vect}(M)$ be the space of smooth vector fields on $M$. This space $V$ is the locally convex topological vector space upon which we are going to base the smooth structure of $\operatorname{Diff}(M)$. Let $\epsilon >0$ be sufficiently small so that all geodesics are uniquely defined by their endpoints (i.e. defines a geodesically convex neighbourhood around each point). Define $$V_0 = \{ (x \mapsto v(x)) \in V: \| v(x) \| < \epsilon, \forall x \in M \}$$ and for each $v \in V_0$ define the map $\phi_v: M \to M$ which takes $p \in M$ to the endpoint of the geodesic segment with initial point $p$, initial velocity $v(x)$, and length $\| v(x) \|$. Our condition on $\epsilon$ guarantees that this is well defined.

This defines a homeomorphism $\phi: V_0 \to U_0\subseteq C^\infty(M,M)$ where the elements of $U_0$ are the $f:M \to M$ sufficiently close to $\operatorname{id}_M$ so that $f(p) \neq -p$. Let $U_1 \subseteq U_0$ be the diffeomorphisms of $U_0$ and $V_1 = \phi^{-1}(U_1)$, so that $\phi^{-1}: U_1 \to V_1$ is a chart of $\operatorname{Diff}(M)$. As $\operatorname{Diff}(M)$ is a group, one can construct an atlas by taking all left-$\operatorname{Diff}(M)$ translates of $U_1$, and this gives the smooth structure.

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    $\begingroup$ Why would this $\epsilon$ exist? I know that it exists locally, but if $M$ is not compact, how do we get a global lower bound? $\endgroup$
    – tomasz
    Commented Jun 5, 2014 at 11:15
  • $\begingroup$ Now, think why your answer is wrong.... $\endgroup$ Commented Feb 19 at 19:40
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See 1.3.2 of the paper, Infinite-dimensional Lie Groups.

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