Besse projective spaces with many diameters

TL;DR

This paper explores the relationship between Besse and Blaschke manifolds, establishing conditions under which Besse manifolds are Blaschke and providing bounds on geodesic lengths for certain curvature metrics.

Contribution

It proves that Besse manifolds with many diameter realizing directions are Blaschke and offers bounds on shortest closed geodesics based on diameter and curvature.

Findings

Besse manifolds with enough diameter directions are Blaschke.

Bounds on shortest closed geodesic length in pinched curvature metrics.

Conditions linking geodesic properties to manifold geometry.

Abstract

It is known that Blaschke manifolds (where injectivity radius equals diameter) are Besse manifolds (where all geodesics are closed). We show that Besse manifolds with sufficiently many diameter realizing directions are Blaschke. We also provide bounds in terms of diameter on the length of the shortest closed geodesic for pinched curvature metrics on simply connected manifolds.