Riemannian Rigidity of the Parallel Postulate in Total Curvature

TL;DR

This paper proves that among Riemannian planes with total curvature and no conjugate points, only the Euclidean plane satisfies Playfair's parallel postulate, highlighting a rigidity property.

Contribution

It establishes a Riemannian rigidity result linking total curvature, conjugate points, and the parallel postulate, characterizing the Euclidean plane uniquely.

Findings

Euclidean plane is unique under the given conditions.

Total curvature and absence of conjugate points imply Euclidean geometry.

Playfair's parallel postulate characterizes Euclidean plane in this setting.

Abstract

We prove that the Euclidean plane is the only Riemannian plane with total curvature and free of conjugate points that satisfies Playfair's version of the parallel postulate.