Positive weighted sectional curvature
Positive weighted sectional curvature
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DOI：
10.1512/iumj.2017.66.6013
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发表时间：
2014-10
期刊：
影响因子：
--
通讯作者：
Lee Kennard;W. Wylie
中科院分区：
文献类型：
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作者：
Lee Kennard;W. Wylie
本文给出了正截面曲率的一种新的推广，即正加权截面曲率。它取决于黎曼度规和光滑向量场的选择。我们给出了几个简单的黎曼度量的例子，这些黎曼度量没有正的截面曲率，但支持一个给它们正加权曲率的向量场。另一方面，我们将具有正截面曲率的紧流形的一些基本结果推广到正加权曲率。特别地，我们证明了Weinstein定理、O'Neill的淹没公式、Frankel定理和Wilking的连通性引理的推广。作为这些结果的应用，我们恢复了高对称阶正曲率流形的Grove-Searle和Wilking拓扑分类结果的加权版本。
In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field that gives them positive weighted curvature. On the other hand, we generalize a number of the foundational results for compact manifolds with positive sectional curvature to positive weighted curvature. In particular, we prove generalizations of Weinstein's theorem, O'Neill's formula for submersions, Frankel's theorem, and Wilking's connectedness lemma. As applications of these results, we recover weighted versions of topological classification results of Grove-Searle and Wilking for manifolds of high symmetry rank and positive curvature.