Characterization of manifolds of constant curvature by spherical curves

TL;DR

This paper establishes that the characterization of geodesic spherical curves via linear equations implies the manifold has constant sectional curvature, providing new geometric criteria and extending results to semi-Riemannian manifolds.

Contribution

It proves the converse characterization of manifolds of constant curvature using properties of geodesic spherical curves and introduces new geometric conditions involving curvature and torsion.

Findings

Geodesic spherical curves characterized by linear equations imply constant sectional curvature.

Small-radius geodesic spheres are totally umbilical in such manifolds.

Results extend to semi-Riemannian manifolds of constant curvature.

Abstract

It is known that the so-called rotation minimizing (RM) frames allow for a simple and elegant characterization of geodesic spherical curves in Euclidean, hyperbolic, and spherical spaces through a certain linear equation involving the coefficients that dictate the RM frame motion (da Silva, da Silva in Mediterr J Math 15:70, 2018). Here, we shall prove the converse, i.e., we show that if all geodesic spherical curves on a Riemannian manifold are characterized by a certain linear equation, then all the geodesic spheres with a sufficiently small radius are totally umbilical and, consequently, the given manifold has constant sectional curvature. We also furnish two other characterizations in terms of (i) an inequality involving the mean curvature of a geodesic sphere and the curvature function of their curves and (ii) the vanishing of the total torsion of closed spherical curves in the case of three-dimensional manifolds. Finally, we also show that the same results are valid for semi-Riemannian manifolds of constant sectional curvature.