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h-projectively equivalent metrics

h-投影等效度量

基本信息

批准号：
226608726
负责人：
Professor Dr. Vladimir Matveev
金额：
--
依托单位：
Lehrstuhl für Differentialgeometrie
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2012
资助国家：
德国
起止时间：
2011-12-31 至 2017-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/226608726?language=en
关键词：
projectively equivalent metrics

项目摘要

Two Kähler metrics on one complex manifold are h-projective equivalent, if they have the same h-planar curves. Such curves are a generalization of geodesics for Kähler manifolds and are defined by the property that the acceleration in every point is complex-proportional to the velocity. The theory of h-projectively equivalent metrics was introduced in the 50thand was actively studied in 60th and 70th. Recently, a group of powerful local and global methods was developed and proved to be effective in the investigation of h-projectively equivalent metrics. The new local methods come from the so called parabolic geometry, which is a popular modern branch of Cartan geometry. The new global methods come in particular from the theory of integrable systems. It was observed that the geodesic flows of nontrivially h-projectively equivalent metrics are Liouville-integrable. One can use this property to obtain topological obstructions for the existence of h-projectively equivalent metrics on compact manifolds. This connection can also work in the other direction: one can use h-projectively equivalent metrics to construct new interesting examples of integrable systems. Very recently, in 2011, it was realized, that h-projectively equivalent metrics were independently introduced and investigated under the name ``Hamiltonian 2-forms''. The methods that were used came from the symplectic, Kähler and toric geometry.We are going to apply these three groups of methods to study the local and global theory of h-projective metrics and h-projective transformations. Specifically, we would like to answer the following questions: 1. Normal form problem: Find local normal forms for h-projectivelyequivalent metrics. 2. Find differential invariants (i.e., invariant algebraic expressions onentries of the metrics and their derivatives) that vanish if and only ifa metric admits nontrivial h-projective equivalence.3. Lie Problem: Find (local) metrics admitting nontrivial pseudogroupof h-projective transformations.4. Prove of disprove the weak topological conjecture: A connectedmanifold admitting two complete nontrivially h-projectively equiva-lent metrics has finite fundamental group. 5. Prove of disprove the strong topological conjecture: A connectedmanifold admitting two complete nontrivially h-projectively equivalent metrics can be decomposed into the product of complex projective spaces and a flat space. 6.Describe all pairs of h-projectively equivalent Einstein (or weakly Bochner-flat, constant scalar curvature, Calabi-extremal,etc.) metrics on closed manifolds.7. Study the integrable systems related to h-projectively equivalent metrics: understand whether the integrals survive in the quantum setting (i.e., if we replace the Hamiltonian by the Laplace operator), try to solve the geodesic equation in special funmctions, and introduce the potential energy in the system).

一个复流形上的两个Kähler度量是h-投射等价的，如果它们具有相同的h-平面曲线。等曲线是一个 Kähler流形的测地线的推广，并通过每个点的加速度与速度成复比例的性质定义。 h-射影等价度量理论于50年代提出，并于60年代和70年代得到了积极的研究。最近，一组强大的局部和全局方法并被证明是有效的，在调查的h-射影等价度量。新的局部方法来自所谓的抛物几何，这是一个流行的现代分支的嘉当几何。新的整体方法特别来自可积系统理论。我们观察到非平凡h-射影等价度量的测地线流是Liouville可积的。利用这个性质可以得到紧致流形上存在h-射影等价度量的拓扑障碍。这种联系也可以在另一个方向上起作用：人们可以使用h-投射等价度量来构造新的有趣的可积系统的例子。最近，在2011年，意识到，h-投影等价度量被独立地引入并在“Hamilton 2-形式”的名称下进行研究。本文所用的方法来自于辛几何、Kähler几何和复曲面几何，我们将利用这三组方法来研究h-投射度量和h-投射变换的局部和整体理论。具体而言，我们想回答以下问题：1。正规形问题：找到h-投影等价度量的局部正规形。2.求微分不变量（即，不变量代数表达式（关于度量及其导数的项），当且仅当度量允许非平凡h-投射等价时为零. Lie问题：找到（局部）度量允许非平凡的伪群的h-投射变换.弱拓扑猜想的反证：一个允许两个完备非平凡h-射影等价度量的连通流形具有有限基本群。 5.强拓扑猜想的反证：一个连通流形如果允许两个完备的非平凡的h-射影等价度量，可以分解为复射影空间与平坦空间的乘积。6.描述所有的h-射影等价爱因斯坦对（或弱Bochner平坦，常数标量曲率，Calabi极值等）封闭流形上的度量。7.研究与h-投射等价度量相关的可积系统：了解积分是否在量子环境中生存（即，如果我们用拉普拉斯算子代替哈密顿量），尝试用特殊的函数解测地线方程，并在系统中引入势能）。