Characterization of curves that lie on a geodesic sphere or on a totally geodesic hypersurface in a hyperbolic space or in a sphere

TL;DR

This paper characterizes curves on geodesic spheres and totally geodesic hypersurfaces in hyperbolic spaces and spheres, extending Euclidean frame methods to Riemannian manifolds of constant curvature.

Contribution

It introduces linear equations to characterize such curves in hyperbolic and spherical geometries, generalizing Euclidean results to Riemannian manifolds.

Findings

Geodesic spherical curves are normal curves via exponential map.

Characterization of geodesic spherical curves by non-homogeneous linear equations.

Curves on totally geodesic hypersurfaces are characterized by homogeneous linear equations.

Abstract

The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a geodesic sphere or totally geodesic hypersurface in a Riemannian manifold of constant curvature. Using that geodesic spherical curves are normal curves, i.e., they are the image of an Euclidean spherical curve under the exponential map, we are able to characterize geodesic spherical curves in hyperbolic spaces and spheres through a non-homogeneous linear equation. Finally, we also show that curves on totally geodesic hypersurfaces, which play the role of hyperplanes in Riemannian geometry, should be characterized by a homogeneous linear equation. In short, our results give interesting and significant similarities between hyperbolic, spherical, and Euclidean geometries.