A curvature form for pseudoconnections

TL;DR

This paper introduces a curvature form for pseudoconnections on vector bundles, establishing conditions under which the associated exterior derivative forms a chain complex or chain 2-complex.

Contribution

It defines a new curvature form for pseudoconnections and characterizes when the exterior derivative forms a chain complex or chain 2-complex.

Findings

The curvature form $F^ abla$ is explicitly constructed.

$F^ abla=0$ is necessary but not sufficient for a chain complex.

$F^ abla=0$ and $d^ abla ext{d}^ abla ext{d}^ abla=0$ are necessary and sufficient for a chain 2-complex.

Abstract

We obtain the curvature form $F^\nabla=P\circ d^\nabla\circ\nabla-d^\nabla\circ P\circ\nabla+d^\nabla\circ\nabla\circ P$ for a vector bundle pseudoconnection $\nabla$, where $d^\nabla$ is the exterior derivative associated to $\nabla$. We use $F^\nabla$ to obtain the curvature of $\nabla$. We also prove that $F^\nabla=0$ is a necessary (but not sufficient) condition for $d^\nabla$ to be a chain complex. Instead we prove that $F^\nabla=0$ and $d^\nabla\circ d^\nabla\circ\nabla=0$ are necessary and sufficient conditions for $d^\nabla$ to be a {\em chain $2$-complex}, i.e., $d^\nabla\circ d^\nabla\circ d^\nabla=0$.