h-projectively equivalent metrics

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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).