Jacobi fields and conjugate points for a projective class of sprays

TL;DR

This paper studies how Jacobi fields and conjugate points behave under projective changes in sprays, providing conditions for their preservation and emergence, with illustrative examples.

Contribution

It establishes that conjugate points are preserved under projective changes and identifies conditions for projectively deformed sprays to have conjugate points.

Findings

Conjugate points are preserved under projective transformations.

Conditions are derived for projectively deformed sprays to have conjugate points.

Illustrative examples demonstrate the theoretical results.

Abstract

We investigate Jacobi fields and conjugate points in the context of sprays. We first prove that the conjugate points of a spray remain preserved under a projective change. Then, we establish conditions on the projective factor so that the projectively deformed spray meets the conditions of a proposition that ensures the existence of conjugate points. We discuss our methods by means of illustrative examples, throughout the paper.