Vector Fields with a non-degenerate Source

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 $Γ^\infty(U, \V)$, 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 $A$.

29 pages

References in corpus (1)

• Asymptotic heat kernel expansion in the semi-classical limit

Cited by in corpus (2)

• Asymptotic eigenfunctions for Schrödinger operators on a vector bundle
• A Semiclassical Heat Kernel Proof of the Poincaré-Hopf Theorem