Curvature Measures in Convex and Integral Geometry
凸几何和积分几何中的曲率测量
基本信息
- 批准号:442235491
- 负责人:
- 金额:--
- 依托单位:
- 依托单位国家:德国
- 项目类别:Research Grants
- 财政年份:
- 资助国家:德国
- 起止时间:
- 项目状态:未结题
- 来源:
- 关键词:
项目摘要
This project builds on decisive progress in Convex and Integral Geometry arising from the work of S. Alesker on valuations, i.e., finitely additive functions on the space of convex bodies. The space of continuous and translation-invariant valuations admits a natural finite grading. By Alesker's Irreducibility Theorem, each graded component is irreducible under the action of the general linear group. This has strong implications for the structure of the space of valuations and led S. Alesker to discover a range of natural algebraic operations on valuations, in particular a commutative product. Furthermore the restriction to the convex setting turned out to be unnatural, and S. Alesker has introduced a theory of smooth valuations on general smooth manifolds.Smooth valuations may be localized, albeit not uniquely, to smooth curvature measures, a crucial new concept introduced by A. Bernig and J.H.G. Fu. In more geometric terms, a smooth curvature measure is an integral of invariants of the second fundamental form that persists under certain singular degenerations. The prime classical examples are Federer's curvature measures, which localize the intrinsic volumes. By Alesker's Irreducibility Theorem, the closure of smooth valuations in the topology of uniform convergence on compact sets is precisely the space of continuous valuations. A fundamental issue and main objective in this project is to similarly complete the space of smooth curvature measures and to characterize the elements of the completion by a short list of inevitable properties. The project will persue the idea that many theorems on translation-invariant valuations should have a counterpart for curvature measures in a few different contexts. In particular, the action of the general linear group will be investigated. Smooth curvature measures also open up a new perspective on the intrinsic volumes. Alesker has shown that the notion of intrinsic volumes from Convex Geometry associates to each Riemannian manifold a finite-dimensional algebra of valuations, the Lipschitz-Killing algebra. The fact that the space of smooth curvature measures is naturally a module over smooth valuations opens up a new and evidently very natural approach: By a conjecture of A. Bernig, J.H.G. Fu, and S. Solanes, a smooth valuation leaves invariant (with respect to the Alesker product) the subspace angular curvature measures, which is distinguished by a simple geometric property, if and only if it is an element of the Lipschitz-Killing algebra. Recently we have established the ``if part'' of this conjecture. One of our primary goals in this project is to explore the ``only if part'' of this conjecture.
这个项目建立在凸和积分几何的决定性进展所产生的工作S。Alesker关于估值,即,凸体空间上的可加函数。连续和连续不变赋值的空间允许自然的有限分次。根据Alesker不可约性定理,每个分次分支在一般线性群的作用下是不可约的。这对估值空间的结构有很强的影响,并导致S。Alesker发现了一系列关于估值的自然代数运算,特别是交换乘积。此外,对凸设置的限制被证明是不自然的,并且S。Alesker提出了一般光滑流形上的光滑赋值理论,光滑赋值可以局部化(尽管不是唯一的)到光滑曲率测度,这是A.伯尼格和JHG Fu.在更多的几何术语中,光滑曲率测度是第二基本形式的不变量的积分,其在某些奇异退化下持续存在。主要的经典例子是费德勒的曲率措施,本地化的内在体积。根据Alesker不可约定理,紧集上一致收敛拓扑中光滑赋值的闭包就是连续赋值空间。该项目的一个基本问题和主要目标是 类似地完成了光滑曲率测度的空间,并通过一系列不可避免的性质来表征完成的元素。该项目将追求的想法,许多定理的曲率不变估值应该有一个对应的曲率措施在一些不同的情况下。特别地,将研究一般线性群的作用。光滑曲率度量也为研究内禀体积开辟了一个新的视角。 阿列斯克已经证明,来自凸几何的内禀体积的概念将每个黎曼流形关联到一个有限维赋值代数,即Lipschitz-Killing代数。 光滑曲率测度的空间自然是光滑赋值上的模这一事实开辟了一个新的、显然非常自然的方法:Bernig,J.H.G. Fu和S. Solanes,一个光滑的赋值使得子空间角曲率测度不变(相对于Alesker乘积),这是由一个简单的几何性质区分的,当且仅当它是Lipschitz-Killing代数的一个元素。最近,我们建立了这个猜想的“如果部分”。 我们在这个项目中的主要目标之一是 来探索这个猜想的“只有当部分”。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Professor Dr. Thomas Wannerer其他文献
Professor Dr. Thomas Wannerer的其他文献
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{{ truncateString('Professor Dr. Thomas Wannerer', 18)}}的其他基金
The Transfer Principle of Integral Geometry and Isoperimetric Inequalities
积分几何传递原理与等周不等式
- 批准号:
289866435 - 财政年份:2016
- 资助金额:
-- - 项目类别:
Research Grants
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