Riemannian Geometry of Loop Spaces

Details Riemannian Geometry of Loop Spaces Riemannian Geometry of Loop Spaces

2007-05-07

arxiv.org/abs/0705.1008v4

Mathematics

Differential Geometry

Cleaner proofs of the main results; new application showing that the fundamental group of Diff(S^2 x S^3) is infinite

Scientific paper

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter. We compute the Levi-Civita connection for integer Sobolev parameter. The connection and curvature forms take values in pseudodifferential operators, and we compute the top symbols of these forms. We develop a theory of Chern-Simons classes in the odd cohomology of LM, using the Wodzicki residue on pseudodifferential operators. We use these "Wodzicki-Chern-Simons" classes to distinguish some circle actions on M = S^2 x S^3, and show that the fundamental group of Diff(M) is infinite.

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