The Weighted Connection and Sectional Curvature for Manifolds With Density
The Weighted Connection and Sectional Curvature for Manifolds With Density
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DOI：
10.1007/s12220-018-0025-3
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发表时间：
2017-07
期刊：
影响因子：
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通讯作者：
Lee Kennard;W. Wylie;Dmytro Yeroshkin
中科院分区：
文献类型：
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作者：
Lee Kennard;W. Wylie;Dmytro Yeroshkin
在本文中，我们从最后两位作者最近引入的加权无扭转连接的角度研究了具有密度的黎曼流形的截面曲率界。我们开发了两种新工具来研究加权截面曲率界限：新的加权劳赫比较定理和距离函数凸性的修改概念。作为应用，我们证明了 Preissman 和 Byers 负曲率定理、（同胚）四分之一收缩球定理和 Cheeger 有限性定理的推广。我们还改进了前两位作者对于正加权截面曲率和对称空间的结果。
In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion-free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger’s finiteness theorem. We also improve results of the first two authors for spaces of positive weighted sectional curvature and symmetry.