Vector Fields with a non-degenerate Source

TL;DR

This paper studies the solution theory of differential operators involving a vector field with a non-degenerate source, establishing Fredholm properties and smooth dependence on data.

Contribution

It provides a detailed analysis of operators of the form ∇_X + A with non-degenerate sources, including Fredholm alternative and smooth dependence results.

Findings

Operators satisfy Fredholm alternative

Solutions depend smoothly on data

Analysis applies to vector bundles with connection

Abstract

We discuss the solution theory of operators of the form $\nabla_X + A$, acting on smooth sections of a vector bundle with connection $\nabla$ over a manifold $M$, where $X$ is a vector field having a critical point with positive linearization at some point $p \in M$. As an operator on a suitable space of smooth sections $\Gamma^\infty(U, \V)$, it fulfills a Fredholm alternative, and the same is true for the adjoint operator. Furthermore, we show that the solutions depend smoothly on the data $\nabla$, $X$ and $A$.