  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.
