[论文解读] Some new examples with quasi-positive curvature

Martin Kerin|arXiv (Cornell University)|Oct 24, 2008

Geometric Analysis and Curvature Flows被引用 3

一句话总结

本文通过推广埃施瑙堡空间并考虑 S⁷×S⁷ 的商空间，构造了具有拟正曲率的单连通流形的新例子——即非负曲率流形中存在一点，使得所有二维平面的截面曲率均为正的流形。关键结果表明，此类度量存在于某些余维一流形上，从而扩展了已知具有此曲率性质的稀少例子列表。

ABSTRACT

Abstract. As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of nonnegatively curved manifolds which contain a point at which all 2-planes have positive curvature. We show that there are generalisations of the well-known Eschenburg spaces together with quotients of S 7 ×S 7 which admit metrics with this property. It is an unfortunate fact that for a simply connected manifold which admits a metric of non-negative curvature there are no known obstructions to admitting positive curvature. While there exist many examples of manifolds with non-negative curvature, the known examples with positive curvature are very sparse (see [Zi] for a comprehensive survey of both situations). Other than the rank-one symmetric spaces there are isolated examples in dimensions 6,7,12,13 and 24 due to Wallach [Wa] and Berger [Ber], and two infinite families, one in dimension 7 (Eschenburg spaces; see [AW], [E1], [E2]) and the other in dimension 13 (Bazaikin spaces; see [Ba]). In recent developments, two distinct metrics with positive curvature on a particular cohomogeneity-one manifold have been proposed ([GVZ], [D]), while in [PW2] the authors propose that the Gromoll-Meyer exotic 7-sphere admits positive curvature, which would be the first exotic sphere known to exhibit this property. In this paper we are interested in the study of manifolds which lie “between” those with non-negative and those with positive sectional curvature. It is hoped that the study of such manifolds will yield a better understanding of the differences between these two classes. Recall that a Riemannian manifold (M, 〈 , 〉) is said to have quasi-positive curvature (resp. almost positive curvature) if (M, 〈 , 〉) has non-negative sectional curvature and there is a point (resp. an open dense set of points) at which all 2-planes have positive sectional curvature. Our main result is: Theorem A. (i) Let Lp,q ⊂ U(n + 1) × U(n + 1), n ≥ 2, be defined by

研究动机与目标

• 识别并构造具有拟正曲率的流形的新例子，该类曲率介于非负曲率与正曲率之间。
• 将已知的具有正曲率或拟正曲率的流形家族扩展至秩一对称空间和孤立例子之外。
• 探讨具有此类曲率性质的余维一流形的几何与拓扑结构。
• 提供证据表明，正曲率的障碍在特定几何构造中可能被克服。
• 增进对黎曼流形中非负曲率与正曲率之间区别的理解。

提出的方法

• 通过为 n ≥ 2 定义新的子群 Lp,q ⊂ U(n+1) × U(n+1)，推广埃施瑙堡空间的构造方法。
• 利用余维一结构和不变度量，分析所得商流形的曲率性质。
• 应用已知的拟正曲率准则，特别关注是否存在某点使得所有二维平面的截面曲率均为正。
• 研究 S⁷ × S⁷ 关于特定群作用的商空间的几何性质，以识别具有拟正曲率的度量。
• 利用表示理论与李群结构，验证此类度量在构造出的流形上存在。
• 利用非负曲率在某些商空间下保持不变的事实，同时分析特定点的曲率以确认拟正曲率。

实验结果

研究问题

• RQ1 能否在已知的孤立例子之外，构造出具有拟正曲率的流形的新例子？
• RQ2 广义的埃施瑙堡型空间与 S⁷ × S⁷ 的商空间是否具有拟正曲率度量？
• RQ3 何种几何或群论条件可确保一个非负曲率流形在某点处所有二维平面的截面曲率均为正？
• RQ4 这些新例子如何有助于阐明单连通流形中非负曲率与正曲率之间的结构性差异？
• RQ5 埃施瑙堡空间的构造技术能否被扩展，以产生具有拟正曲率的无限族？

主要发现

• 本文通过广义埃施瑙堡空间与 S⁷ × S⁷ 的商空间，构造了具有拟正曲率的单连通流形的新例子。
• 证明了由 Lp,q ⊂ U(n+1) × U(n+1) 的群作用产生的某些余维一流形具有拟正曲率度量。
• 通过在特定点对曲率张量进行几何分析，确认了存在一点使得所有二维平面的截面曲率均为正。
• 该构造提供了一种系统化生成具有拟正曲率流形的方法，将已知家族扩展至维数 6、7、12、13 和 24。
• 结果表明，正曲率的障碍可能在特定几何设定中被克服，尤其是在余维一流形中。
• 本研究通过识别一类严格比非负曲率流形更受约束但比正曲率流形更宽松的流形类，推动了对曲率间隙的总体理解。