Maslov index in semi-Riemannian submersions

TL;DR

This paper investigates the Maslov index and focal points of horizontal geodesics in semi-Riemannian submersions, establishing explicit relations with base space geodesics and applying results to stationary space-times.

Contribution

It provides an explicit relation between focal points and Maslov indices of geodesics in total and base spaces of semi-Riemannian submersions, with applications to stationary space-times.

Findings

Derived explicit formulas relating focal points and Maslov indices in semi-Riemannian submersions.

Calculated the focal Maslov index for spacelike geodesics in stationary space-times.

Connected geometric properties of total space geodesics with their projections in the base space.

Abstract

We study focal points and Maslov index of a horizontal geodesic $\gamma:I\to M$ in the total space of a semi-Riemannian submersion $\pi:M\to B$ by determining an explicit relation with the corresponding objects along the projected geodesic $\pi\circ\gamma:I\to B$ in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary space-time which is orthogonal to a timelike Killing vector field.