Singularities of the eigenvalue functions for first order symmetric symbols on rank two vector bundles over surfaces

TL;DR

This paper investigates the singularities of eigenvalue functions for symmetric symbols on rank two bundles over surfaces, linking their existence and degree to topological invariants and optical phenomena.

Contribution

It introduces a topological framework for understanding eigenvalue singularities and derives a formula for their degree in terms of Euler characteristics.

Findings

Singularities of eigenvalue functions are topologically determined.

The degree of singularities relates to optical phenomena directions.

A formula connects the degree to Euler characteristics of the surface and associated manifolds.

Abstract

For a rank two bundle $F$ over a surface $X$, we study the set of singularities $M_\sigma$ of the eigenvalue functions of symmetric symbols $\sigma$ associated to first order differential operators on $F$. We prove that the existence of these singularities follows from topological considerations. We define the degree of the set of singularities and show that it can be used to count the number of directions at which special optical phenomena occur. For the case when $F=TX$ we compute a formula for the degree in terms of the Euler characteristics of $X$ and $N_\sigma$, where $N_\sigma\subset X$ is a manifold with boundary associated to $\sigma$.