Raffaelli Matteo
Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8-10/104, 1040 Vienna, Austria.
Geom Dedic. 2023;217(6):96. doi: 10.1007/s10711-023-00833-8. Epub 2023 Aug 23.
A curve in a Riemannian manifold is if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when lies on an oriented hypersurface of , we say that is if the curve's principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that is three-dimensional and closed. We show that if is a well-positioned line of curvature of , then its total torsion is an integer multiple of ; and that, conversely, if the total torsion of is an integer multiple of , then there exists an oriented hypersurface of in which is a well-positioned line of curvature. Moreover, under the same assumptions, we prove that the total torsion of vanishes when is convex. This extends the classical total torsion theorem for spherical curves.
黎曼流形中的一条曲线是[具体条件未给出，原文此处有缺失]，如果它的挠率（带符号的二阶曲率函数）是定义良好的且所有高阶曲率都恒等于零。特别地，当[曲线所在流形相关条件缺失]位于[流形中某个有向超曲面相关条件缺失]的一个有向超曲面上时，我们说[曲线相关条件缺失]是[具体条件未给出，原文此处有缺失]，如果曲线的主法线、其挠率向量和曲面法线处处共面。假设[流形相关条件缺失]是三维且封闭的。我们证明，如果[曲线相关条件缺失]是[流形相关条件缺失]的一条位置良好的曲率线，那么它的总挠率是[具体数值未给出，原文此处有缺失]的整数倍；反之，如果[曲线相关条件缺失]的总挠率是[具体数值未给出，原文此处有缺失]的整数倍，那么存在[流形相关条件缺失]的一个有向超曲面，在其中[曲线相关条件缺失]是一条位置良好的曲率线。此外，在相同假设下，我们证明当[曲线相关条件缺失]是凸的时候，[曲线相关条件缺失]的总挠率为零。这扩展了关于球面曲线的经典总挠率定理。
