AMS Mathematics Subject Classification Numbers: 53C05, 58J40. RIEMANNIAN GEOMETRY ON LOOP SPACES

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter s. In Part I, we compute the Levi-Civita connection for these metrics for s ∈ Z. The connection and curvature forms take values in pseudodifferential operators (ΨDOs), and we compute the top symbols of these forms. In Part II, we develop a theory of Wodzicki-Chern-Simons classes CS K ∈ H2k−1(LM2k−1), for K = (k1, ..., k`) a partition of 2k−1, using the Wodzicki residue on ΨDOs. We use CS 5 to prove that certain actions on S×S are not smoothly homotopic, and that π1(Diff(S×S)) is infinite.