General Notion of Curvature in Catastrophe Theory Terms

TL;DR

This paper introduces a novel curvature concept for superconnections, linking classical geometric obstructions to a unified framework inspired by catastrophe theory, enhancing understanding of singularities in smooth sections of vector bundles.

Contribution

It proposes a new curvature notion for superconnections that generalizes classical obstructions and unifies various curvature-related concepts under a single approach.

Findings

New curvature notion for superconnections introduced

Classical obstructions interpreted via the new framework

Unified approach to curvature and singularities developed

Abstract

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.