  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 singularities of smooth sections of
vector bundles (Catastrophe Theory). We demonstrate that the classical examples
of obstructions to a local equivalence: exterior differential for 2-forms,
Riemannian tensor, Weil tensor, curvature of a linear connection and Nijenhuis
tensor can be treated in terms of one general approach. This approach, applied
to the superconnection leads to a new notion of a curvature (proposed in this
paper) of a superconnection.
