A curvature form for pseudoconnections

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., $d^{\nabla }\circ d^{\nabla }\circ d^{\nabla }=0$.

6 pages