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The general notion of a curvature in catastrophe theory terms. (English) Zbl 1192.53027
Mladenov, Ivaïlo M. (ed.), Proceedings of the 9th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 8–13, 2007. Sofia: Bulgarian Academy of Sciences (ISBN 978-954-8495-42-4/pbk). 265-279 (2008).
Summary: We introduce a new notion of a curvature of a superconnection, different from the one obtained by a purely algebraic analogy with the curvature of a linear connection. The naturalness of this new notion of curvature of a superconnection comes from the study of the singularities of smooth sections of vector bundles (catastrophe theory). We demonstrate that the classical examples of obstructions to a local equivalence: exterior differential for two-forms, Riemannian tensor, Weil tensor, curvature of a linear connection and Nijenhuis tensor can be treated in terms of some general approach. This approach, applied to the superconnection leads to the new notion of curvature (proposed in the paper) of a superconnection.
MSC:
53C07Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills)
58C50Analysis on supermanifolds or graded manifolds
58K99Theory of singularities and catastrophe theory