group of diffeomorphisms of a manifold

Question

How much has been the group of diffeomorphisms of a manifold " been studied. I got this information from wiki. " Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the Witt algebra, which has a central extension called the Virasoro algebra, used in string theory and conformal field theory. Very little is known about the diffeomorphism groups of manifolds of larger dimension. The diffeomorphism group of spacetime sometimes appears in attempts to quantize gravity."

.What more is known about this?Has this group been calculated for the standard manifolds.Since this group is a big group,so what are the better ways of studying this object.

Answer 1

One can approach the study of diffeomorphism groups from many perspectives: topology, geometry, differential equations, and dynamics. I'll mention a few results that I'm aware of, giving links to literature surveys on different topics.

There is a short exact sequence $$Diff_0(M)\to Diff(M)\to MCG(M),$$ where $Diff_0(M)$ is the subgroup of diffeomorphisms isotopic to the identity.

One can regard $MCG(M)=\pi_0(Diff(M))$. There is a huge literature studying $MCG(M)$, especially when $M$ is a surface. One question that has been answered for closed surfaces is that there is no section $Diff(M)\leftarrow MCG(M)$. I'm not sure what's known about the higher-dimensional version of this question.

Topologists study the homotopy type of $Diff(M)$, which breaks down into computing $MCG(M)$ and the homotopy type of $Diff_0(M)$. Hatcher has a survey on the homotopy type of $Diff(M)$. This has more-or-less been completely resolved in dimensions $\leq 3$, but is quite complex for general higher dimensional manifolds.

It is known that $Diff_0(M)$ is simple for closed manifolds by a result of Thurston. A general strategy then for understanding the group structure of $Diff_0(M)$ is to understand its subgroups. One aspect of this is the Zimmer program, to understand homomorphisms $\Lambda\to Diff_0(M)$, where $\Lambda$ is a higher rank lattice. Another aspect is to consider homomorphisms between diffeomorphism groups for different manifolds.

There are some results on dynamics of diffeomorphisms with relation to the diffeomorphism group. There is a huge literature on the dynamics of individual diffeomorphisms, but I think this is orthogonal to your question.

Answer 2

In addition to the references suggested in the above comments, I'd recommend taking a look at recent work of B. Khesin, D. Ebin, G. Misiolek, S. Preston, P. Michor among others... B. Khesin has a nice book (freely available on his webpage) about infinite-dimensional groups, with a whole chapter on Diffeomorphism groups, that is perhaps a good place to start.

Just for a glimpse of what goes on beyond "just" topology, exotic structures, etc., the geometry of a few interesting subgroups of the diffeomorphism group $\mathcal D^s(M)$ of a manifold $M$ is also an important object of study. For example, take $\mathcal D_\mu^s(M)$, formed by volume preserving diffeomorphisms. This group is very much related with classical equations of hydrodynamics: you can think of the motion of an incompressible fluid filling a manifold $M$ as a curve in $\mathcal D_\mu^s(M)$. Classic work of Arnold (1966) and Ebin & Marsden (Ann. of Math., 1970), later followed the above mentioned authors, establishes a very beautiful setup for the Euler equations $\partial_t u+\nabla_u u=-\nabla p$, $div\ u=0$ (and something similar can be done for Navier-Stokes), proving that solutions $u(t,x)$ to these PDEs on $M$ are the $1$-parameter families of volume preserving diffeomorphisms that arise as geodesics in $\mathcal D_\mu^s(M)$ for an $L^2$ Riemannian metric in this infinite-dimensional manifold. This approach also works for some other PDEs, like Burgers' equation and KdV, the latter having to do with the diffeomorphism group of the circle. Most of this is discussed in detail in Khesin's book.

In this way, weak Riemannian geometry of the diffeomorphism group of $M$ and some of its submanifolds is deeply interconnected with many evolution equations on $M$. Of course, as the OP mentions, low dimension plays a big role in having more answers for now. For example, one can prove global existence of solutions to some of these PDEs of hydrodynamics in 2D using the above setup, but the 3D version is open and worth some big prizes and big money.

It is also worth pointing out that, recently, the infinite-dimensional geometry of diffeomorphism groups has been related to areas other than hydrodynamics, like optimal transport and geometric statistics, see this paper.

Answer 3

Weak right invariant Riemannian metrics on full diffeomorphism groups have vanishing geodesic distance if the Sobolev order of the metric is smaller (or equal in the the case of $S^1$) than 1/2. This is true for the metrics where the geodesic equation corresponds to the KdV equation also (here one needs the Virasoro group), and the Burgers equation. See the following paper and references therein.

Martin Bauer, Martins Bruveris, Peter W. Michor: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II. 7 pages. arXiv:1211.7254. (pdf)