Geometric Submanifolds of Manifolds

流形的几何子流形

• 批准号: 1105710
• 负责人: David McReynolds
• 金额: $ 14.6万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105710&HistoricalAwards=false
• 关键词: Geometric; Submanifolds; Manifolds

This project is centers around understanding Riemannian manifolds via submanifolds. This topic and these methods have been central in Riemannian geometry since its conception, and have direct ties to problems and methods in geometric group theory. With regard to submanifolds, the PI seeks to further investigate the geometric data encoded by the primitive totally geodesic submanifolds of a fixed Riemannian n-manifold. The simplest case of 1-dimensional totally geodesic submanifolds is nothing more than the geodesic length spectrum and has garnered interest for more than 50 years. In addition, the geodesic length spectrum has strong connections with the eigenvalue spectrum of the Laplace-Beltrami operator and analysis on manifolds. In addition to this study, the PI plans on investigating which manifolds arise as totally geodesic submanifolds of a fixed Riemannian manifold. Two cases are locally symmetric manifolds of non-compact type and the moduli space of genus g curves with n marked points. In the latter case, most prominent are the Teichmuller curves and their associated fundamental groups Veech groups, which are interesting geometrically, algebro-geometrically, and dynamically. In addition, the PI seeks obstructions or special properties of submanifolds of a fixed manifold, not only in an effort to understand the fixed manifold but also to better understand obstructions to isometrically immersing manifolds into other manifolds. Lastly, the PI plans to investigate the asymptotic behavior of geometric counting functions, what geometric data is encoded by these functions, and ties to this topic outside of geometry. These function counting the totally geodesic manifolds of a fixed type as a function of volume and a basic entities, which in the simplest cases are known to have ties to geometric dynamics and applications to number theory.The objects that arise in the present proposal permeate mathematics as fundamental examples tied to basic problems and areas. They were fathered not by mathematics but from pure and applied science. Moreover, over time, have been proven to be centrally important not only in mathematics but in physics, chemistry, and computer science. The PI hopes to foster further these important connections not only directly with specific results but also in disseminating the core ideaology of the PI's proposal.

这个项目的中心是通过子流形来理解黎曼流形。这一主题和这些方法从黎曼几何的概念开始就一直是它的中心，并与几何群论中的问题和方法有直接的联系。对子流形，PI试图进一步研究固定黎曼n-流形的本原全测地子流形所编码的几何数据。一维全测地子流形最简单的情况就是测地线长度谱，50多年来一直受到人们的关注。此外，测地线长度谱与Laplace-Beltrami算子的本征值谱和流形上的分析有很强的联系。除了这项研究之外，PI还计划研究哪些流形是固定黎曼流形的全测地子流形。两种情形是局部对称的非紧型流形和具有n个标记点的亏格g曲线的模空间。在后一种情况下，最突出的是TeichMuller曲线及其相关的基本群Veech群，它们在几何、代数几何和动力学上都很有趣。此外，PI寻找固定流形的子流形的障碍或特殊性质，不仅是为了理解固定流形，也是为了更好地理解等距浸入其他流形的障碍。最后，PI计划研究几何计数函数的渐近行为，这些函数编码了哪些几何数据，并与几何之外的主题联系在一起。这些函数将固定类型的全测地流形计数为体积和基本实体的函数，在最简单的情况下，已知这些实体与几何动力学和数论的应用有关。在本提案中出现的目的渗透到数学中，作为与基本问题和领域相关的基本例子。他们不是数学之父，而是纯粹的应用科学之父。此外，随着时间的推移，它们不仅在数学中，而且在物理、化学和计算机科学中都被证明是核心重要的。国际和平协会希望进一步促进这些重要的联系，不仅直接取得具体成果，而且还传播国际和平协会提案的核心理念。