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Characterizing round spheres using half-geodesics.

Ian M Adelstein¹, Benjamin Schmidt²

¹Department of Mathematics, Yale University, New Haven, CT 06520; ian.adelstein@yale.edu.

Proceedings of the National Academy of Sciences of the United States of America

|July 3, 2019

Summary

This summary is machine-generated.

This study shows that Riemannian spheres with closed geodesics and many half-geodesics are round. This confirms that specific geodesic properties define a sphere

Keywords:

Blaschke manifolds Zoll spheres closed geodesics

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Area of Science:

Differential Geometry
Riemannian Geometry
Topology

Background:

Geodesics are fundamental paths in Riemannian manifolds.
Half-geodesics are closed geodesics that realize the distance between any two points.
Round spheres have all geodesics as half-geodesics.

Purpose of the Study:

To investigate the characterization of round spheres using geodesic properties.
To determine if the converse of a known property of round spheres holds true.
To establish conditions under which a Riemannian sphere is necessarily round.

Main Methods:

Analysis of geodesic behavior in Riemannian manifolds.
Utilizing the definition of half-geodesics.
Applying geometric and topological arguments.

Main Results:

Demonstration that Riemannian spheres with all closed geodesics and sufficiently many half-geodesics are round.
Establishment of a converse to the property that all geodesics in a round sphere are half-geodesics.

Conclusions:

The property of having all closed geodesics and sufficiently many half-geodesics characterizes round spheres.
This research provides a deeper understanding of the relationship between geodesics and the geometry of spheres.