Spaces with sectional and Ricci curvature bounds

具有截面和里奇曲率边界的空间

• 批准号: RGPIN-2017-06369
• 负责人: Kapovitch, Vitali
• 金额: $ 1.75万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679989
• 关键词: Spaces; sectional; Ricci; curvature; bounds

Curvature of a space is its most important local geometric invariant. It describes how a space is shaped or curved locally from the point of view of an observer living inside that space. ***This proposal is mostly concerned with spaces with various lower curvature bounds. The notions of spaces with lower sectional and Ricci curvature bounds play an important role in a number of fields in mathematics and physics: general relativity, optimal transport, big data analysis and, of course, Riemannian geometry itself. I plan to study several problems concerning generalized spaces with lower sectional curvature bounds (Alexandrov spaces) and generalized spaces with lower Ricci curvature bounds (Restricted Curvature-Dimension condition). I am particularly interested in the interplay between these two conditions which as yet is not sufficiently understood.

曲率是空间最重要的局部几何不变量。 它描述了一个空间是如何形成或弯曲的局部从一个观察者的角度生活在该空间。* 这个建议主要涉及具有各种曲率下界的空间。具有较低截面和里奇曲率界限的空间概念在数学和物理的许多领域发挥着重要作用：广义相对论、最优传输、大数据分析，当然还有黎曼几何本身。本文主要研究具有下截曲率界的广义空间（Alexandrov空间）和具有下Ricci曲率界的广义空间（限制曲率-维数条件）的若干问题。我特别感兴趣的是这两个条件之间的相互作用，这还没有得到充分的理解。