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Let $M$ be a compact manifold and $\text{diff}(M)$ its diffeomorphism group. Various quotients of $\text{diff}(M)$ appear in the literature, oftentimes with geometric significance. A well-known example is the space $\text{diff}(M)/\text{diff}_\mu(M)$, where $\mu$ is a measure on $M$ of total volume one and $\text{diff}_\mu(M)$ is the group of diffeomorphisms of $M$ that preserve $\mu$. (See Hamilton's Nash-Moser paper for more details.)

Questions:

(1) Is $\text{diff}(M)/\text{diff}_\mu(M)$ a tame Fréchet manifold?

(2) Is $\text{diff}_\mu(M)$ a tame Fréchet submanifold of $\text{diff}(M)$? (Hamilton's paper says that this is an open problem. Has any progress been made on this since 1982?)

(3) If $G$ is a subgroup of the diffeomorphism group, when is $\text{diff}(M)/G$ a tame Fréchet manifold?

I know only of a result that addresses (3) on totally different lines. This is the Gleason-Yamabe theorem. It says that if $G$ is locally compact, then for any open neighborhood $U$ of the identity there exists a subgroup $G'$ of $G$ contained in $U$ and a compact $G'$-normal subgroup $K\subset U$ such that $G'/K$ is isomorphic to a Lie group. (Terence Tao has written about this theorem in connection with Hilbert's Fifth Problem.) Since I want to quotient out the full diffeomorphism group by a subgroup, this won't do.

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1 Answer 1

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(1) Yes, it is the set of smooth positive densities of the same total volume as $\mu$. This can be seen by the "Moser trick" directly, and it is a one dimensional affine subspace of a space of smooth post ice sections of a line bundle, that tame.

(2) Yes, it is, see

  • MR2670086 (2011m:58006) Reviewed Molitor, Mathieu(CH-LSNP-SM) The group of unimodular automorphisms of a principal bundle and the Euler-Yang-Mills equations. (English summary) Differential Geom. Appl. 28 (2010), no. 5, 543–564.

for a version in a more complicated situation. The original version is in Molitor's thesis.

(3) If $G$ is a tamely splitting submanifold, then one can use the Hamilton-Nash-Moser implicit function theorem. If $G$ is the isotropy group of some geometric object on which $Diff(M)$ acts, often one can identify the the quotient.

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