Maslov, Chern-Weil and Mean Curvature

Tommaso Pacini

Published 2017-11-21Version 1

We provide an integral formula for the Maslov index of a pair $(E,F)$ over a surface $\Sigma$, where $E\rightarrow\Sigma$ is a complex vector bundle and $F\subset E_{|\partial\Sigma}$ is a totally real subbundle. As in Chern-Weil theory, this formula is written in terms of the curvature of $E$ plus a boundary contribution. When $(E,F)$ is obtained via an immersion of $(\Sigma,\partial\Sigma)$ into a pair $(M,L)$ where $M$ is K\"ahler and $L$ is totally real, the boundary contribution is related to the mean curvature of $L$. In this context the formula thus allows us to control the Maslov index in terms of the geometry of $(M,L)$.