Characterization of manifolds of constant curvature by spherical curves
Characterization of manifolds of constant curvature by spherical curves
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DOI：
10.1007/s10231-019-00874-5
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发表时间：
2020-02-01
影响因子：
1
通讯作者：
da Silva, Jose D.
中科院分区：
文献类型：
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作者：
da Silva, Luiz C. B.;da Silva, Jose D.
众所周知，所谓的旋转最小化（RM）框架允许通过涉及决定RM框架运动的系数的特定线性方程来简单而优雅地表征欧几里得，双曲和球面空间中的测地线球面曲线（da Silva，da Silva in Mediterr J Math 15：70，2018）。在这里，我们将证明匡威的情况，即，我们证明，如果黎曼流形上的所有测地球面曲线都由某个线性方程刻画，那么所有半径足够小的测地球面都是全脐的，因此，给定的流形具有恒定的截面曲率。我们还提供了另外两个方面的特征：（i）不等式涉及的平均曲率的测地球和曲率函数的曲线和（ii）消失的情况下，三维流形的闭球面曲线的总挠率。最后，我们还证明了同样的结果也适用于常截面曲率的半黎曼流形。
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.