arXiv research

In 2D, Finsler metrics with 3+ projective fields are projectively equivalent to Randers.

The paper studies projectively equivalent para-Kaehler metrics in 4D.

Generalizes quantum integrability to all signatures for projectively equivalent metrics.

Trajectories of light rays in a static spacetime are described by unparametrised geodesics of the Riemannian optical metric associated with the Lorentzian spacetime metric. We investigate the uniqueness of this structure and demonstrate that two different observers, moving relative to one another, who both see the univ…

We show that $PQ^ε$-projectivity of two Riemannian metrics introduced in \cite{Top2003} implies affine equivalence of the metrics unless $ε\in\{0,-1,-3,-5,-7,...\}$. Moreover, we show that for $ε=0$, $PQ^ε$-projectivity implies projective equivalence.

Scaling and 1-form addition make projectively equivalent Finsler metrics.

Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.

Geodesic flows on Kähler manifolds are quantum integrable when metrics are c-projectively equivalent.

Two Kaehler metrics on one complex manifold are said to be c-projectively equivalent if their J-planar curves, i.e., curves defined by the property that their acceleration is complex proportional to their velocity, coincide. The degree of mobility of a Kaehler metric is the dimension of the space of metrics that are c-…

Two Kähler metrics on a complex manifold are called c-projectively equivalent if their $J$-planar curves coincide. These curves are defined by the property that the acceleration is complex proportional to the velocity. We give an explicit local description of all pairs of c-projectively equivalent Kähler metrics of arb…

Two pseudo-Riemannian metrics are called projectively equivalent if their unparametrized geodesics coincide. The degree of mobility of a metric is the dimension of the space of metrics that are projectively equivalent to it. We give a complete list of possible values for the degree of mobility of Riemannian and Lorentz…

Consider a smooth manifold $M$ equipped with a bracket generating distribution $D$. Two sub-Riemannian metrics on $(M,D)$ are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric $g$ is called rigid …

This paper combines two classical theories, namely metric projective differential geometry and superintegrability. We study superintegrable systems on 2-dimensional geometries that share the same geodesics, viewed as unparametrized curves. We give a definition of projective equivalence of such systems, which may be con…

We correct a mistake in Shen Yibing, Yu Yaoyong, On Projectively Related Randers Metrics, International Journal of Mathematics 19}(2008), no. 5, 503--520, and prove the natural generalization of the projective Lichnerowicz-Obata conjecture for Randers metrics.

It is well known that pseudo-Riemannian metrics in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the solutions of a certain overdetermined projectively invariant differential equation. This equation is a special case of a so-called first BGG equation. The ge…

The degree of mobility of a (pseudo-Riemannian) Kähler metric is the dimension of the space of metrics h-projectively equivalent to it. We prove that a metric on a closed connected manifold can not have the degree of mobility $\ge 3$ unless it is essentially the Fubini-Study metric, or the h-projective equivalence is a…

We construct local normal forms of pseudo-Riemannian projectively equivalent 2-dimensional metrics.

The note proves a metric equivalence for stable bundles on surfaces.

Describes geodesic scattering on hyperboloids using quadrics results.

With the help of a generalization of the Fermat principle in general relativity, we show that chains in CR geometry are geodesics of a certain Kropina metric constructed from the CR structure. We study the projective equivalence of Kropina metrics and show that if the kernel distributions of the corresponding 1-forms a…

Undergrad project: Shows geodesics coincide in Heisenberg group under two metrics.

The Bochner tensor is the Kähler analogue of the conformal Weyl tensor. In this article, we derive local (i.e., in a neighbourhood of almost every point) normal forms for a (pseudo-)Kähler manifold with vanishing Bochner tensor. The description is pined down to a new class of symmetric spaces which we describe in terms…

A trivial projective change of a Finsler metric $F$ is the Finsler metric $F + df$. I explain when it is possible to make a given Finsler metric both forward and backward complete by a trivial projective change. The problem actually came from lorentz geometry and mathematical relativity: it was observed that it is poss…

The study connects norms and filtrations on section rings of projective manifolds.

In Theorem 1, we generalize the results of Szabo for Berwald metrics that are not necessary strictly convex: we show that for every Berwald metric F there always exists a Riemannian metric affine equivalent to F. As an application we show (Corollary 3) that every Berwald projectively flat metric is a Minkowski metric; …

A vector field on a Kähler manifold is called c-projective if its flow preserves the J-planar curves. We give a complete local classification of Kähler real 4-dimensional manifolds that admit an essential c-projective vector field. An important technical step is a local description of 4-dimensional c-projectively equiv…

A metric projective structure is a manifold equipped with the unparametrised geodesics of some pseudo-Riemannian metric. We make acomprehensive treatment of such structures in the case that there is a projective Weyl curvature nullity condition. The analysis is simplified by a fundamental and canonical 2-tensor invaria…

We show that, on the connected sum of complex projective planes, any toric LeBrun metric can be identified with a Joyce metric admitting a semi-free circle action through an explicit conformal equivalence. A crucial ingredient of the proof is an explicit connection form for toric LeBrun metrics.

This paper explores new Finsler metrics and their connections to generalized Sakaguchi's Theorem.

Uniform bounds prove connection between Kähler metrics and RCD spaces.

Two metrics $g$ and $\bar g$ are geodesically equivalent, if they share the same (unparameterized) geodesics. We introduce two constructions that allow one to reduce many natural problems related to geodesically equivalent metrics, such as the classification of local normal forms and the Lie problem (the description o…

Study of 2D metrics with one projective symmetry leading to superintegrable systems.

We classify all Kahler metrics in an open subset of $C^2$ whose real geodesics are circles. All such metrics are equivalent (via complex projective transformations) to Fubini metrics (i.e. to Fubini-Study metric on $CP^2$ restricted to an affine chart, to the complex hyperbolic metric in the unit ball model or to the E…

The paper is related to the classification of special manifolds and projective special manifolds. One of the result of this paper is that, if the Weil-Petersson metric on a projective special manifold is complete, then the Hodge metrc and the Weil-Petersson metrc are equivalent.

In this paper, we prove the equivalence of the existence of extremal Kahler metrics and the properness of the modified K energy on projective bundles. Moreover, we discuss the relations of the lower boundedness of the K energy, the infimum of the Calabi energy and the extremal polynomials. In particular, we give an exa…

New equivalence found for flat vector bundles without extra conditions.

In this paper, we prove that if a compact Kähler manifold $X$ has a smooth Hermitian metric $ω$ such that $(T_X,ω)$ is uniformly RC-positive, then $X$ is projective and rationally connected. Conversely, we show that, if a projective manifold $X$ is rationally connected, then the tautological line bundle $\mathscr{O}_{T…

Projective pre-compactness induces projective structure on boundary, relates to GR asymptotic forms.

The paper is devoted to the investigation of four-dimensional Kahler manifolds admitting non-affine H-projective mappings. We find all such manifolds which are non-Einstein. In the paper also Kahler manifolds admitting infinitesimal H-projective transformations are determined. It is proved that the class of Kahler mani…

We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral curves of a vector field on the tangent bundle: the geodesic vector field assoc…

In this paper known results of symmetric orthogonality, as introduced by G. Birkhoff, and non-expansive nearest point projections are extended from the linear to the metric setting. If the space has non-positive curvature in the sense Busemann then it is shown that those concepts are actually equivalent. In the end it …

Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.

Study on manifolds with higher-rank lattice actions in projective and conformal classes.

Study maps in semidirect products of groups, proving Lipschitz properties without intrinsic dilations.

The paper studies new curvature properties in Finsler geometry.

Study of quasi-Kähler metrics on complex manifolds linked to c-projective metrizability.

The mobility of a Kaehler metric is the dimension of the space of metrics with which it is c-projectively equivalent. The mobility is at least two if and only if the Kaehler metric admits a nontrivial hamiltonian 2-form. After summarizing this relationship, we present necessary conditions for a Kaehler metric to have m…

For complete affine manifolds we introduce a definition of compactification based on the projective differential geometry (i.e.\ geodesic path data) of the given connection. The definition of projective compactness involves a real parameter $α$ called the order of projective compactness. For volume preserving connectio…