Characterization of manifolds of constant curvature by spherical curves

TY - JOUR

T1 - Characterization of manifolds of constant curvature by spherical curves

AU - da Silva, Luiz C. B.

AU - da Silva, José D.

N1 - © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

PY - 2020/2

Y1 - 2020/2

N2 - 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.

AB - 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.

KW - Rotation minimizing frame

KW - Totally umbilical submanifold

KW - Geodesic sphere

KW - Spherical curve

KW - Space form

UR - https://www.scopus.com/pages/publications/85067678354

U2 - 10.1007/s10231-019-00874-5

DO - 10.1007/s10231-019-00874-5

M3 - Article

SN - 1618-1891

VL - 199

SP - 217

EP - 229

JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

ER -