Geodesics in 3-dimensional Euclidean Space with One or Two Analytic Obstacles

TL;DR

This paper proves that in three-dimensional Euclidean space with certain analytic obstacles, the number of components of shortest paths is finite and independent of initial velocity, extending previous results to more complex boundary configurations.

Contribution

It establishes the finiteness and velocity-independence of geodesic components in 3D with multiple analytic obstacles, generalizing prior work to intersecting boundary surfaces.

Findings

Number of geodesic components is finite in 3D with obstacles.

Finiteness of components is independent of initial velocity.

Results extend to regions with intersecting analytic hypersurfaces.

Abstract

In many singular metric spaces, the regularity of a shortest-length curve is unknown. Algebraic varieties, or more generally sets defined by finitely many polynomial or real analytic equalities or inequalities, all locally partition into finitely many real analytic submanifolds called strata. So any component of a shortest-length curve which lies completely in one such stratum is a geodesic in the stratum, hence an embedded real analytic curve. The key question thus is whether there are only finitely many components. F. Albrecht and I.D. Berg proved this is true for a geodesic in a closed region of $n$-dimensional Euclidean space with a smooth real analytic hyper surface as boundary. Here the curve consists of finitely many interior line segments alternating with boundary hypersurface geodesics. Their bound on the number of these depended on the initial velocity of the geodesic, and they conjectured that is independent. Here we prove this independence in $\mathbb{R}^3$. We also generalize their result to regions whose boundary is locally the boundary of the union of two transversally intersecting analytic hypersurfaces in $\mathbb{R}^3$.