Minimal Orbits of Metrics and Elliptic Operators

The group of diffeomorphisms of a compact manifold acts isometrically on the space of Riemannian metrics with its L 2 metric. Following [1], [15], we define minimal orbits for this action by a zeta function regularization. We show that odd dimensional isotropy irreducible homogeneous spaces give rise to minimal orbits, and find a flat two torus giving a stable minimal orbit. We also define an infinite dimensional family of elliptic operators on a bundle over a manifold M with an action by automorphisms of the bundle. The orbits are parametrized by the metrics on M . In odd dimensions, all orbits are minimal if the cohomology of the elliptic complex vanishes. In this case, the determinant of an associated elliptic operator is a smooth invariant of M . This invariant is defined for some classes of 3-manifolds, including the simply connected ones. It is similar to analytic torsion, and has a combinatorial analogue. 1 Partially supported by the JSPS and BMWF. 2 Partially supported by ..