Thursday, 26th November 2009, 4:00 pm

Mathematics Building, Lecture Hall 2

David Ebin

(Stony Brook)

"Geodesics on Groups of Diffeomorphisms"

** Abstract: **

We begin by constructing a Banach manifold structure on the set of
maps from a compact manifold *M* to another manifold *N*.
Given Riemannian metrics on the two manifolds, we can find geodesics
on this space of maps. If *M = N*, we can look at the group of
diffeomorphisms, *Diff*, of *M*, which is an open subset
of this space of maps.

Next we look at *SDiff*, the diffeomorphisms that preserve the volume
element of *M*, a subgroup and submanifold of *Diff*.
Using the submanifold structure and its 2nd fundamental form, we find
geodesics on *SDiff*. These correspond to motions
of an incompressible inviscid fluid in *M*.

Finally we assume that *M* has a symplectic form and using it we define
*SymDiff*, the group of symplectic diffeomorphisms.
We construct geodesics on this group as well.

In the case that *M* is two-dimensional, *SDiff* and *SymDiff*
coincide and we can show geodesic completeness.
Geodesic completeness for SymDiff in the higher dimensional case is work
in progress.

The work on SymDiff is new, but the other work goes back over several decades and involves a number of authors.

Light refreshments will be served in the faculty lounge at 3:30.

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