Geometry of Riemannian and Finsler Spaces

黎曼空间和芬斯勒空间的几何

• 批准号: 9803129
• 负责人: Dmitri Burago
• 金额: $ 14.46万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803129&HistoricalAwards=false
• 关键词: Geometry; Riemannian; Finsler; Spaces

Abstract Proposal: DMS-9803129 Principal Investigator: Dmitry Burago The main topics of the proposal (large-scale geometry and billiards systems) belong to both geometry and dynamical systems. Continuing his (joint with S. Ivanov) research in the large-scale geometry of periodic metrics, D. Burago proposes to study the asymptotic volume growth for Finsler tori; a related problem is to analyze area-minimizing properties of affine subspaces in Banach spaces. This may help to understand how the absence of conjugate points for a Lagrangian system on a torus is reflected in the dynamics of its geodesic flow. D. Burago also proposes to continue his analysis of higher dimensional analogs for geometric conclusions of Aubry-Mather theory. Trying to further understand the relationship between biLipschitz equivalence and quasi-isometries, it is natural to consider such cases where density arguments (developed by the proposer jointly with B. Kleiner) do not work; the most striking of such cases include Penrose tilings and uniform lattices. Continuing his (joint with S. Ferleger, A. Kononenko) study of semi-dispersing billiard systems, D. Burago plans to investigate if their topological entropy can be infinite. Another problem which arose from the proposer's method of applying singular geometry to billiard theory is constructing CAT(0) development spaces whose geodesics represent all billiard trajectories. E. Johnson, the proposer's advisee, works on applying the proposer's method to prove the unboundness for complete surfaces of finite variation of curvature to show stability of the class of embedded flat surfaces. The main subjects of the proposal have very clear physical analogs. Large-scale geometric properties of periodic metrics can be interpreted as global properties of a periodic medium consisting of copies of the same microscopic pattern repeated in a regular fashion, as in crystals. In particular, dynamical properties of geo desic flows for periodic metrics reflect how the light or radiation spreads in such media. The technique developed by the proposer and his collaborators allows us to solve many of the open problems in this area for the most important case (quadratic Lagrangians); in general, this circle of problems remains wide open. The proposer's research in the theory of billiard systems started from a problem that goes back to Boltzmann: can one give an upper bound on the maximum number of collisions in a given time interval in a system of several balls colliding elastically (gas model)? The number of collisions per time unit plays important role in thermodynamics. Surprisingly, this problem has been solved by establishing a connection with singular geometry; in its turn, this connection led to new intriguing problems of both geometric and dynamical origin.

摘要 提案：DMS-9803129主要研究者：Dmitry Burago 该提案的主要议题（大型几何和台球 系统）属于几何和动力系统。 继续 他（与S。伊万诺夫）大规模几何研究 periodic metrics，D. Burago建议研究渐近体积 Finsler环面的增长;一个相关的问题是分析 Banach空间中仿射子空间的面积极小性质。 这 可能有助于理解如何缺乏共枕点的一个 环面上的拉格朗日系统反映在其动力学中， 测地线流 D. Burago还建议继续分析 Aubry-Mather几何结论的高维类比 理论 为了进一步了解 biLipschitz等价和拟等距，这是自然的， 考虑这样的情况，其中密度参数（由提议者开发 与B联合。Kleiner）不起作用;此类案件中最引人注目的是 包括彭罗斯镶嵌和均匀晶格。 继续他的（联合） 链球菌Ferleger，A. Kononenko）半分散台球的研究 systems，D. Burago计划研究它们的拓扑熵 可以是无限的。 另一个问题是， 一种将奇异几何应用于台球理论的方法是 构造CAT（0）发展空间，其测地线表示所有 台球轨迹 e.约翰逊，提议者的代理人， 应用proposer方法证明了完全 曲率的有限变化的表面，以显示 嵌入平面的一类。 该提案的主要主题有非常明确的物理类似物。 周期度量的大尺度几何性质可以 解释为周期性介质的全局属性， 以有规律的方式重复相同的微观图案的副本， 就像水晶一样。 特别是，geo desic流的动力学性质 因为周期性的度量反映了光或辐射如何传播 这样的媒体。 由提议者和他的 合作者使我们能够解决许多开放的问题， 最重要情况下的面积（二次拉格朗日）;一般来说， 这一系列问题仍然没有得到解决。 提案人的研究 台球系统的理论起源于一个问题 玻尔兹曼：一个人能给一个上限的最大数量的 在多个球的系统中在给定时间间隔内的碰撞 弹性碰撞（气体模型）？ 每次碰撞的次数 单位在热力学中起着重要的作用。 令人惊讶的是，这 通过建立与奇异值的连接，问题已得到解决 几何学;反过来，这种联系导致了新的有趣的问题 几何学和动力学的起源。