Symmetry in Solvmanifolds and Geometric Evolutions

求解流形和几何演化中的对称性

• 批准号: 1105647
• 负责人: Michael Jablonski
• 金额: $ 10.6万
• 依托单位: University of Oklahoma Norman Campus
• 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: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105647&HistoricalAwards=false
• 关键词: Symmetry; Solvmanifolds; Geometric; Evolutions

In this project, the principal investigator proposes to develop a deeper understanding of isometry groups of homogeneous spaces by evolving Riemannian and Lie structures to look for highly symmetric Riemannian metrics on a given homogeneous space. The hybrid techniques used lie at the intersection of Riemannian geometry and Geometric Invariant Theory and are motivated by geometric evolutions like the Ricci flow. In the presence of a transitive nilpotent group of isometries, the proposed techniques have been successfully employed by the principal investigator to show that Ricci soliton metrics on nilmanifolds have maximal isometry groups. The principal investigator proposes to develop this approach in the more general setting that the transitive group of isometries is solvable to achieve similar results for Einstein and Ricci soliton metrics on solvmanifolds. Similar questions on compact nilmanifolds will also be addressed. As several of the proprosed problems can naturally be rephrased in the language of Geometric Invariant Theory, this avenue will be explored as well.This project is devoted to the fundamental problem in geometry of finding a best shape for a given object. The objects of interest are homogeneous spaces, which are spaces having the property that every point looks the same as every other point. These spaces serve as basic examples across many branches of mathematics and have been a source of inspiration in modern geometry for over a century. Although homogeneous spaces have been tirelessly explored, there are still many fundamental questions that have not been resolved. The principal investigator will address this question of finding preferred metrics, aiming to show that Einstein and Ricci solitons metrics are the most symmetric choice of geometry on a fixed homogeneous space, when they exist.The PI will continue his work, supervising undergraduate research and attracting graduate students for area of research. The project has the potential to provide attractive and challenging opportunities for the undergraduate and graduate students of his university through the study of the Ricci flow on solvable Lie groups. The PI will continue his work, supervising undergraduate research and attracting graduate students for his area of research. The project has the potential to provide attractive and challenging opportunities for the undergraduate and graduate students of his university through the study of the Ricci flow on solvable Lie groups.

在这个项目中，主要研究者提出通过发展黎曼和李结构来更深入地理解齐性空间的等距群，以寻找给定齐性空间上的高度对称黎曼度量。 所使用的混合技术位于黎曼几何和几何不变理论的交叉点，并受到像Ricci流这样的几何演化的激励。 在存在可迁幂零等距群的情况下，主要研究者成功地利用所提出的方法证明了幂零流形上的Ricci孤子度量具有极大等距群. 主要研究者提出，发展这种方法在更一般的设置，传递组的等距是可解的，以实现类似的结果，爱因斯坦和里奇孤子度量的solvmanifold。 紧nilmanifold类似的问题也将得到解决。 由于几个proprosed问题可以很自然地在语言中的几何不变理论，这条途径也将探讨。这个项目是致力于在几何的基本问题，找到一个最佳形状为给定的对象。 感兴趣的对象是齐次空间，这些空间具有每个点看起来与其他每个点相同的属性。 这些空间作为数学许多分支的基本例子，并且在世纪以来一直是现代几何的灵感来源。 虽然均匀空间已经被不知疲倦地探索，但仍然有许多基本问题没有解决。 首席研究员将解决这个问题，找到首选的度量，旨在表明爱因斯坦和里奇孤子度量是最对称的选择几何上一个固定的齐次空间，当他们存在。PI将继续他的工作，监督本科生的研究，并吸引研究生的研究领域。该项目有可能通过研究可解李群上的Ricci流，为他所在大学的本科生和研究生提供有吸引力和具有挑战性的机会。PI将继续他的工作，监督本科生的研究，并吸引研究生为他的研究领域。该项目有可能通过研究可解李群上的Ricci流，为他所在大学的本科生和研究生提供有吸引力和具有挑战性的机会。