Intrinsic and extrinsic integral geometry

摘要奖：DMS-9972094主要研究员：约瑟夫·H·G·傅我们建议研究某些奇异空间的几何，这些空间的曲率自然定义为测度。所考虑的曲线是在Steiner-Weyl管公式中出现的那些曲线。对于光滑空间，这些量是曲率积分。对于奇异空间，它们是某些可视为广义曲率的符号测度的全变差。在所有已知的情况下，这些曲率度量是关于欧氏空间中奇异空间的嵌入的先验定义，实际上是从空间的固有度量结构计算出来的。我们的总体目标是理解和推广这一现象。这个问题是亚历山大和费德勒在20世纪50年代单独攻击的。亚历山德罗夫的方法是内在的，但仅限于二维，而费德勒的理论处理的是一般维度的嵌入物体。进一步的进展将取决于这两种方法的协调。我们提出，技术关键是基于电流的几何测量理论。本文的出发点是，费德勒的理论可以被重塑为：对于欧几里德空间中的某些奇异子空间X，在环境空间的球丛中存在相关联的积分流N(X)。这个流满足勒让德条件(即它消灭了由正则1-形式生成的微分理想)，并由它与X的Morse理论的关系决定。然后，X的曲率度量可以从N(X)以正则的方式计算。我们的建议是基于该建议的最新工作，使用当前的理论框架，表明亚历山大的理论可以用类似的术语来理解。这暗示了将内在方法推广到更高维度的可能性。简单地说，这个项目的目标是开发一种数学语言，足以描述我们每天遇到的几何形状：例如，一张纸或岩石表面的碎屑。尽管自然界中常见的几何形状往往是粗糙的(“奇异的”)，但由于技术上的原因，几何在历史上的发展限制了它对光滑物体的关注。我们的目标是能够以数学的方式处理这样的物体，而不强加这样的现实假设。我们预计，这项工作将在计算机可视化和板材屈曲研究(如碰撞过程中的挡泥板)等领域产生技术影响。

AbstractAward: DMS-9972094Principal Investigator: Joseph H.G. FuWe propose to study the geometry of certain singular spaces,whose curvatures are naturally defined as measures. Thecurvatures under consideration are those arising in theSteiner-Weyl tube formula. For smooth spaces these quantitiesare curvature integrals. For singular spaces they are the totalvariations of certain signed measures which may be regarded asgeneralized curvatures. In all known cases these curvaturemeasures, which are a priori defined with respect to an embeddingof the singular space in Euclidean space, are in fact computablesolely from the intrinsic metric structure of the space. Ourgeneral goal is to understand and extend this phenomenon. Thisproblem was attacked independently by Alexandrov and by Federerin the 1950s. Alexandrov's approach was intrinsic but limited to2 dimensions, while Federer's theory dealt with embedded objectsof general dimension. Further progress will depend on thereconciliation of these two methods. We propose that thetechnical key is based on the use of the geometric measure theoryof currents. The starting point is the fact that Federer'stheory may be recast in the following terms: to certain singularsubspaces X in euclidean space there is an associated integralcurrent N(X) in the sphere bundle of the ambient space. Thiscurrent satisfies the Legendre condition (i.e. it annihilatesthe differential ideal generated by the canonical 1-form) and isdetermined by its relation with the Morse theory of X. Thecurvature measures of X are then computable in a canonical wayfrom N(X). Our proposal is based on recent work of the proposerusing the current theoretic framework, showing that Alexandrov'stheory may be understood in similar terms. This suggests thepossibility of generalizing the intrinsic approach to higherdimensions.In ordinary terms, this project aims to develop a mathematicallanguage adequate to describe geometrical forms we encounterevery day: for example, the crumplings of a sheet of paper or thesurface of a rock. Even though geometrical shapes typicallyfound in nature tend to be rough ("singular"), for technicalreasons geometry as it has developed historically restricts itsattention to smooth objects. Our goal is to be able to dealmathematically with such objects without imposing suchunrealistic assumptions. We expect that this work willeventually have a technological impact, in such areas as computervisualization and the study of buckling of sheets (as of a carfender during a collision).