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The group of diffeomorphisms of a manifold M can be
given
a differentiable structure which
makes it in many ways like an infinite-dimensional Lie group.
Symplectic diffeomorphisms form a sub-group and
also a sub-manifold. A Riemannian structure on M induces one on
the
group of diffeomorphisms and therefore on the
sub-group of symplectic diffeomorphisms. This structure gives an
equation for geodesics on the subgroup which is a system of
partial
differential
equations. It turns out that this system can be expressed as an
ordinary differential equation on a function space and therefore
can
be solved
by Picard iteration. Thus one can construct geodesics and get an
exponential map.
In addition using right invariance of the metric together with some rather sensitive estimates on the flow of a vector field, one can show that geodesics extend without bound, or that the sub-group is geodesically complete. The whole construction is similar (one might even say analogous) to the case of the subgroup of volume-preserving diffeomorphisms. |