D’Atri空间和半球、管和圆柱体的总标量曲率。

IF 1.4 3区 数学 Q1 MATHEMATICS
Revista Matematica Complutense Pub Date : 2023-01-01 Epub Date: 2022-10-10 DOI:10.1007/s13163-022-00444-z
Balázs Csikós, Amr Elnashar, Márton Horváth
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引用次数: 0

摘要

Csikós和Horváth在J Geom Anal 28(4):3458-3476,(2018)中证明,如果维度至少为4的连通黎曼流形是调和的,那么小半径管关于任意正则曲线的总标量曲率仅取决于曲线的长度和管的半径,反之,如果后一个条件适用于圆柱体,即。,对于关于测地线段的管,则流形是调和的。在本文中,我们证明了与高维情况相反,连通的三维黎曼流形具有上述管的性质,当且仅当该流形是D’Atri空间,此外,如果该空间具有有界截面曲率,则仅对圆柱体要求总标量曲率条件就足以暗示该空间是D’Atri。这一结果否定了盖森和万赫克提出的问题。为了证明这些陈述,我们给出了D’Atri空间在任何维度上测地半球的总标量曲率方面的特征。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
D'Atri spaces and the total scalar curvature of hemispheres, tubes and cylinders.

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 geodesic 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.

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来源期刊
CiteScore
2.30
自引率
0.00%
发文量
28
审稿时长
>12 weeks
期刊介绍: Revista Matemática Complutense is an international research journal supported by the School of Mathematics at Complutense University in Madrid. It publishes high quality research and survey articles across pure and applied mathematics. Fields of interests include: analysis, differential equations and applications, geometry, topology, algebra, statistics, computer sciences and astronomy. This broad interest is reflected in our interdisciplinary editorial board which is comprised of over 30 internationally esteemed researchers in diverse areas. The Editorial Board of Revista Matemática Complutense organizes the “Santaló Lecture”, a yearly event where a distinguished mathematician is invited to present a lecture at Complutense University and contribute to the journal. Past lecturers include: Charles T.C. Wall, Jack K. Hale, Hans Triebel, Marcelo Viana, Narayanswamy Balakrishnan, Nigel Kalton, Alfio Quarteroni, David E. Edmunds, Giuseppe Buttazzo, Juan L. Vázquez, Eduard Feireisl, Nigel Hitchin, Lajos Horváth, Hélène Esnault, Luigi Ambrosio, Ignacio Cirac and Bernd Sturmfels. The Santaló Lecturer for 2019 will be Noel Cressie from National Institute for Applied Statistics Research Australia (NIASRA), University of Wollongong.
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