Two remarks on $PQ^{\epsilon}$-projectivity of Riemannian metrics

TL;DR

This paper investigates the conditions under which $PQ^{ ext{epsilon}}$-projectivity of Riemannian metrics implies affine or projective equivalence, revealing exceptions for specific epsilon values.

Contribution

It establishes that $PQ^{ ext{epsilon}}$-projectivity generally implies affine equivalence, except for certain epsilon values, and clarifies the case when epsilon equals zero.

Findings

$PQ^{ ext{epsilon}}$-projectivity implies affine equivalence unless epsilon is in {0, -1, -3, -5, -7, ...}

For epsilon=0, $PQ^{ ext{epsilon}}$-projectivity implies projective equivalence

Identifies specific epsilon values where the implication does not hold

Abstract

We show that $PQ^{\epsilon}$-projectivity of two Riemannian metrics introduced in \cite{Top2003} implies affine equivalence of the metrics unless $\epsilon\in\{0,-1,-3,-5,-7,...\}$. Moreover, we show that for $\epsilon=0$, $PQ^{\epsilon}$-projectivity implies projective equivalence.