h-projectively equivalent metrics

一个复流形上的两个Kähler度量如果具有相同的h-平面曲线，则它们是h-射影等价的。这种曲线是Kähler流形测地线的推广，由每个点的加速度与速度成复正比这一性质来定义。H-射影等价度量理论在50年代被引入，并在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被发展起来，并被证明是研究h-射影等价度量的有效方法。新的局部方法来自所谓的抛物线几何，它是Cartan几何的一个流行的现代分支。新的整体方法尤其来自于可积系统理论。观察到非平凡h-射影等价度量的测地线流是Liouville可积的。利用这一性质可以得到紧致流形上h-射影等价度量存在的拓扑障碍。这种联系也可以在另一个方向上起作用：人们可以使用h-射影等价度量来构造新的有趣的可积系统例子。最近，也就是在2011年，人们意识到h-射影等价度量是独立地引入和研究的，命名为“哈密尔顿2-形式”。所使用的方法来自辛几何、Kähler几何和环面几何，我们将应用这三组方法来研究h-射影度量和h-射影变换的局部和整体理论。具体地说，我们想回答以下问题：1.范式问题：寻找h-射影等价度量的局部范式。2.找到当且仅当IFA度量具有非平凡h-投射等价时消失的微分不变量(即度量及其导数项上的不变代数表达式)。Lie问题：寻找允许h-射影变换的非平凡伪群的(局部)度量。弱拓扑猜想的证明：允许两个完全非平凡h-射影等价度量的连通流形有有限个基本群。5.证明并反驳了强拓扑猜想：具有两个完全非平凡h-投射等价度量的连通流形可以分解为复射影空间和平坦空间的乘积。6.描述所有h-射影等价的爱因斯坦对(或弱Bochner平坦、常标量曲率、Calabi-极值等)。闭流形上的度量。研究与h-射影等价度量有关的可积系统：了解积分在量子环境下是否存在(即，如果我们用拉普拉斯算符代替哈密顿量)，尝试在特殊函数下解测地方程，并引入系统中的势能)。

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).