The azimuthal equidistant projection for a Finsler manifold by the exponential map

Nobuhiro Innami, Yoe Itokawa, Toshiki Kondo, Tetsuya Nagano and Katsuhiro Shiohama

Vol. 308 (2020), No. 1, 73–101

DOI: 10.2140/pjm.2020.308.73

Abstract

Let $\left(M,F\right)$ be a geodesically forward complete Finsler manifold and $p\in M$. We observe how the preimage of a curve in $M$ under exponential map at $p$ can behave in the tangent space $T_{p}M$ at $p$, when the curve approaches a conjugate cut point of $p$ without crossing the cut locus of $p$. After this investigation, we may regard the internal region of a tangent cut locus of $p\in M$ as the development of $M$. We deal with isometry problems of Finsler manifolds and differentiability conditions of cut loci.

Keywords

Finsler manifold, cut locus, azimuthal equidistant projection, exponential map

Mathematical Subject Classification 2010

Primary: 53C20

Secondary: 53C22

Milestones

Received: 4 November 2019

Accepted: 27 May 2020

Published: 3 December 2020