叶状黎曼流形上广义Bott联络诱导的曲率与黎曼曲率的关系
The Relationship between the Curvature Induced by the Generalized Bott Connection and the Riemannian Curvature on the Riemannian Foliations
DOI: 10.12677/AAM.2024.136283, PDF, 下载: 59  浏览: 85 
作者: 史凯悦：浙江师范大学，数学科学学院，浙江 金华
关键词: 叶状黎曼流形；广义Bott联络；曲率关系；Ricci曲率；Riemannian Foliations； Generalized Bott Connection； Curvature Relation； Ricci Curvature
摘要: 本文研究在叶状黎曼流形上, 由广义Bott联络所诱导的曲率与黎曼曲率的关系。进一步推出在广义Bott联络下，当S = 0 时, 流形具有正Ricci 曲率。
Abstract: In this paper, we study the relationship between the curvature induced by the gener- alized Bott connection and the Riemannian curvature on the Riemannian foliations. Furthermore, it is concluded that under the generalized Bott connection, when S = 0, the manifold has positive Ricci curvature.
文章引用：史凯悦. 叶状黎曼流形上广义Bott联络诱导的曲率与黎曼曲率的关系[J]. 应用数学进展, 2024, 13(6): 2966-2974. https://doi.org/10.12677/AAM.2024.136283

参考文献

[1] Lohkamp, J. (1994) Metrics of Negative Ricci Curvature. The Annals of Mathematics, 140, 655-683.
https://doi.org/10.2307/2118620
[2] Gromoll, D. and Meyer, W.T. (1985) Examples of Complete Manifolds with Positive Ricci Curvature. Journal of Differential Geometry, 21, 195-211.
https://doi.org/10.4310/jdg/1214439562
[3] Perelman, G. (1994) Manifolds of Positive Ricci Curvature with Almost Maximal Volume. Journal of the American Mathematical Society, 7, 299-305.
https://doi.org/10.1090/s0894-0347-1994-1231690-7
[4] Cheeger, J. and Gromoll, D. (1971) The Splitting Theorem for Manifolds of Nonnegative Ricci Curvature. Journal of Differential Geometry, 6, 119-128.
https://doi.org/10.4310/jdg/1214430220
[5] Hamilton, R.S. (1982) Three-manifolds with Positive Ricci Curvature. Journal of Differential Geometry, 17, 255-306.
https://doi.org/10.4310/jdg/1214436922
[6] Schoen, R. and Yau, S. (2019) Positive Scalar Curvature and Minimal Hypersurface Singular- ities. Surveys in Differential Geometry, 24, 441-480.
https://doi.org/10.4310/sdg.2019.v24.n1.a10
[7] Zhu, S. (1993) A Finiteness Theorem for Ricci Curvature in Dimension Three. Journal of Differential Geometry, 37, 711-727.
https://doi.org/10.4310/jdg/1214453906
[8] Menguy, X. (2000) Noncollapsing Examples with Positive Ricci Curvature and Infinite Topo- logical Type. Geometric and Functional Analysis, 10, 600-627.
https://doi.org/10.1007/pl00001632
[9] Gromoll, D. and Walschap, G. (2009) Metric Foliations and Curvature. In: Progress in Math- ematics, Vol. 268, Birkhäuser Verlag.
[10] Baudoin, F. and Bonnefont, M. (2015) Curvature-Dimension Estimates for the Laplace- Beltrami Operator of a Totally Geodesic Foliation. Nonlinear Analysis, 126, 159-169.
https://doi.org/10.1016/j.na.2015.06.025
[11] Baudoin, F. (2016) Sub-Laplacians and Hypoelliptic Operators on Totally Geodesic Rieman- nian Foliations. In: Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds, Vol. 1, European Mathematical Society, 259-321.

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