D'Atri spaces and the total scalar curvature of hemispheres, tubes and cylinders.

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Accounts of Chemical Research 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

Abstract

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.

D’Atri空间和半球、管和圆柱体的总标量曲率。
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。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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