# Characterizing Round Spheres Using Half-Geodesics

## TL;DR

This paper characterizes round spheres by their geodesic properties, showing that spheres with all geodesics closed and many half-geodesics are necessarily round, thus linking geodesic behavior to sphere shape.

## Contribution

It proves that Riemannian spheres with all geodesics closed and numerous half-geodesics must be round, providing a new characterization of round spheres based on geodesic properties.

## Key findings

- All geodesics in a round sphere are half-geodesics.
- Spheres with all geodesics closed and sufficiently many half-geodesics are round.
- The paper establishes a converse characterization of round spheres.

## Abstract

A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round.