The General Notion of a Curvature in Catastrophe Theory Terms∗

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 a 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 a new notion of a curvature (proposed in the paper) of a superconnection. 1. A Brief Review of the Notion of a Superconnection The notions of a superconnection and of the corresponding supercurvature were introduced by Quillen in 1985 [7]. In this section we give a brief review of the matter and introduce the basic notations. By ξ = (E, p,M) we denote a vector bundle over the manifold M (dimM = m, dim(ξ) = n), by ξ∗ – the dual bundle and byC∞(ξ) – the space of the vector fields, i.e., the space of the smooth sections of the bundle ξ. Respectively Ωk(M) = C∞(ΛkT ∗(M)) is the the space of the differential k-forms on the manifoldM and