Riemannian Geometry on Loop Spaces
نویسنده
چکیده
A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter. In Part I, we compute the Levi-Civita connection for these metrics. 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 Chern-Simons classes CS 2k−1 ∈ H(LM,R), using the Wodzicki residue on ΨDOs. By results in Part I, for stably parallelizable manifolds these “Wodzicki-Chern-Simons” classes are defined for all metrics and are smooth invariants of M for k > 2. We produce examples of nontrivial three dimensional Wodzicki-Chern-Simons classes.
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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-Che...
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