The exponential map at a cuspidal singularity

TL;DR

This paper investigates the geometry of spaces with cuspidal singularities, focusing on the exponential map at the singularity, revealing its complex behavior and dependence on boundary invariants.

Contribution

It introduces a detailed analysis of the exponential map at cuspidal singularities, including its potential discontinuities and invariants, extending understanding of singular Riemannian geometries.

Findings

Exponential map at cuspidal singularities can be surjective but discontinuous.

Behavior of the exponential map is governed by an invariant function on the link.

Results apply to manifolds with boundary with cuspidal metrics.

Abstract

We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to naturally define an exponential map based at the singularity, but that the behavior of this map can deviate strongly from the behavior of the exponential map based at a smooth point or at a conical singularity: While it is always surjective near the singularity, it may be discontinuous and non-injective on any neighborhood of the singularity. The precise behavior of the exponential map is determined by a function on the link of the singularity which is an invariant -- essentially the only boundary invariant -- of the induced metric. Our methods are based on the Hamiltonian system of geodesic differential equations and on techniques of singular analysis. The results are proved in the more general natural setting of manifolds with boundary carrying a so-called cuspidal metric. Revision: restructured many parts of the paper completely; stated main theorems for cusps of any order; replaced 'cusp' by 'cuspidal', also in title, added references