Minimum $L^\infty$ Accelerations in Riemannian Manifolds

TL;DR

This paper investigates curves in Riemannian manifolds that minimize the maximum acceleration, extending the concept of Riemannian cubics which traditionally minimize the integral of squared acceleration.

Contribution

It introduces necessary conditions for curves minimizing the $L^ extinfty$ norm of acceleration, applicable to spheres and bi-invariant Lie groups, expanding the theory beyond the $L^2$ case.

Findings

Derived necessary conditions for $L^ extinfty$ acceleration minimizers.

Analyzed specific cases on spheres and bi-invariant Lie groups.

Extended understanding of optimal curves in Riemannian geometry.

Abstract

Riemannian cubics are critical points for the $L^2$ norm of acceleration of curves in Riemannian manifolds $M$. In the present paper the $L^\infty$ norm replaces the $L^2$ norm, and a less direct argument is used to derive necessary conditions analogous to those for Riemannian cubics. The necessary conditions are examined when $M$ is a sphere or a bi-invariant Lie group.