Csikós Balázs, Elnashar Amr, Horváth Márton
Department of Geometry, Eötvös Loránd University, Budapest, Hungary.
Central European University, Budapest, Hungary.
Rev Mat Complut. 2023;36(3):887-898. doi: 10.1007/s13163-022-00444-z. Epub 2022 Oct 10.
Csikós and Horváth proved in J Geom Anal 28(4): 3458-3476, (2018) that if a connected Riemannian manifold of dimension at least 4 is harmonic, then the total scalar curvatures of tubes of small radius about an arbitrary regular curve depend only on the length of the curve and the radius of the tube, and conversely, if the latter condition holds for cylinders, i.e., for tubes about segments, then the manifold is harmonic. In the present paper, we show that in contrast to the higher dimensional case, a connected 3-dimensional Riemannian manifold has the above mentioned property of tubes if and only if the manifold is a D'Atri space, furthermore, if the space has bounded sectional curvature, then it is enough to require the total scalar curvature condition just for cylinders to imply that the space is D'Atri. This result gives a negative answer to a question posed by Gheysens and Vanhecke. To prove these statements, we give a characterization of D'Atri spaces in terms of the total scalar curvature of geodesic hemispheres in any dimension.
奇科斯和霍瓦特在《几何分析杂志》28(4): 3458 - 3476, (2018)中证明，如果一个维数至少为4的连通黎曼流形是调和的，那么围绕任意正则曲线的小半径管的总标量曲率仅取决于曲线的长度和管的半径，反之，如果对于圆柱（即围绕线段的管）后一个条件成立，那么该流形是调和的。在本文中，我们表明与高维情形不同，一个连通的三维黎曼流形当且仅当该流形是达特里空间时才具有上述管的性质，此外，如果该空间具有有界截面曲率，那么仅要求圆柱的总标量曲率条件就足以意味着该空间是达特里空间。这个结果对盖森斯和范赫克提出的一个问题给出了否定答案。为了证明这些陈述，我们根据任意维数的测地半球的总标量曲率给出了达特里空间的一个特征描述。
