Not every conjugate point of a semi-Riemannian geodesic is a bifurcation point

TL;DR

This paper clarifies the relationship between conjugate points and bifurcation points in semi-Riemannian geodesics, correcting previous misconceptions and providing improved examples to support the main claim.

Contribution

The paper corrects a previous mistake and demonstrates that not all conjugate points are bifurcation points, offering new examples to substantiate this claim.

Findings

Not every conjugate point is a bifurcation point in semi-Riemannian geodesics

A correction of a previous example shows all conjugate points can be bifurcation points

An improved example confirms the main claim in the title

Abstract

We revisit an example of a semi-Riemannian geodesic that was discussed by Musso, Pejsachowicz and Portaluri in 2007 to show that not every conjugate point is a bifurcation point. We point out a mistake in their argument, showing that on this geodesic actually every conjugate point is a bifurcation point. Finally, we provide an improved example which yields that the claim in our title is nevertheless true.