Warped product metrics on hyperbolic and complex hyperbolic manifolds

TL;DR

This paper derives curvature formulas for warped-product metrics on hyperbolic and complex hyperbolic manifolds with submanifolds removed, facilitating future geometric analysis and applications.

Contribution

It provides explicit curvature formulas for warped-product metrics on hyperbolic and complex hyperbolic manifolds minus totally geodesic submanifolds.

Findings

Curvature formulas expressed in spherical coordinates

Application potential for geometric analysis

Framework for future research in hyperbolic geometry

Abstract

In this paper we study warped-product metrics on manifolds of the form $X \setminus Y$, where $X$ denotes either $\mathbb{H}^n$ or $\mathbb{C} \mathbb{H}^n$, and $Y$ is a totally geodesic submanifold with arbitrary codimension. The main results that we prove are curvature formulas for these metrics on $X \setminus Y$ expressed in spherical coordinates about $Y$. We also discuss past and potential future applications of these formulas.