Nonnegative Curvature on Lie Groups and Bundles

李群和丛上的非负曲率

• 批准号: 0902942
• 负责人: Kristopher Tapp
• 金额: $ 7.87万
• 依托单位: St Joseph's University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0902942&HistoricalAwards=false
• 关键词: Nonnegative; Curvature; Lie; Groups; Bundles

Although tremendous progress has been made towards understanding relationships between curvature and topology, the classical topic of positive/nonnegative sectional curvature still suffers from an insufficient supply of examples. The PI proposes two different methods for exploring rigidity and constructing new examples of Riemannian manifolds with nonnegative and positive sectional curvature. The first method begins with the question: classify the left-invariant metrics with nonnegative curvature on each compact Lie group. Almost all known constructions of nonnegative curvature begin with a bi-invariant metric, so answering this question would help us measure the rigidity of the known constructions, and could lead to new examples of nonnegatively curved manifolds with (at least points of) positive curvature. The second method is related to past research in which the PI developed conditions under which vector bundles admit nonnegative curvature. More precisely, if a vector bundle admits nonnegative curvature, then a fundamental differential inequality relates the three curvatures which are visible at points of its soul: the curvature of the soul, the curvature of the connection in the normal bundle of the soul, and the curvature of planes orthogonal to the soul. Conversely, if a vector bundle admits structure which strictly satisfy this inequality, then the unit sphere bundle admits positive sectional curvature. The PI proposes to use this theorem to find new examples, obstructions, and rigidity theorems for metrics with nonnegative (respectively positive) curvature on vector bundles (respectively sphere bundles).Differential Geometry provides the mathematical language for precisely describing Einstein's theory of relativity, particle physics, and high dimensional curved shapes (called manifolds). The PI's proposed work within this field involves the study of manifolds with positive curvature, which is a visually natural restriction on the way in which a manifold curves about in space. Past examples come from Lie groups, which are indispensable tools in diverse fields of mathematics, physics, cosmology, computer animation, and other disciplines in which simplification is achieved through symmetry. The search for new examples of manifolds with positive curvature has a long history, yet frustratingly few examples have been found. The PI proposes substantially new methods for constructing more examples, which could thereby lead to a better general understanding of the relationship between curvature and global shape.

虽然在理解曲率和拓扑之间的关系方面已经取得了巨大的进展，但正/非负截面曲率的经典主题仍然缺乏例子。PI提出了两种不同的方法来探索刚性和构造具有非负和正截面曲率的黎曼流形的新例子。第一种方法从问题开始：对每个紧李群上具有非负曲率的左不变度量进行分类。几乎所有已知的非负曲率结构都以双不变度量开始，因此回答这个问题将有助于我们衡量已知结构的刚性，并可能导致具有(至少是正曲率的)非负曲线流形的新例子。第二种方法与过去的研究有关，在这些研究中，PI给出了向量丛允许非负曲率的条件。更准确地说，如果向量丛允许非负曲率，那么一个基本的微分不等式将在其灵魂的点上可见的三个曲率联系在一起：灵魂的曲率，灵魂法丛中的连接曲率，以及与灵魂垂直的平面的曲率。反之，如果一个向量丛允许严格满足这一不等式的结构，则单位球丛允许正截曲率。PI建议使用这个定理来寻找向量丛(分别是球丛)上具有非负(分别为正)曲率的度量的新例子、障碍和刚性定理。微分几何为精确描述爱因斯坦的相对论、粒子物理和高维弯曲形状(称为流形)提供了数学语言。PI在这一领域提出的工作涉及研究具有正曲率的流形，这是流形在空间中曲线移动的视觉上的自然限制。过去的例子来自李群，在数学、物理、宇宙学、计算机动画和其他通过对称实现简化的不同领域中，李群是不可或缺的工具。寻找具有正曲率的流形的新例子已经有很长的历史了，但令人沮丧的是，很少有例子被发现。PI提出了构建更多示例的新方法，从而可以更好地全面理解曲率和全局形状之间的关系。