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.Comment: 15 pages, 3 figures; comments are welcom
