arXiv research

Positive projectively flat metrics on Hopf manifolds are locally conformally flat-Kähler.

The paper characterizes complex Finsler metrics that are projectively flat or dually flat.

Study cylindrical symmetric Finsler metrics that are projectively flat.

The paper explores conditions for projective and dually flat metrics in Finsler spaces with Kropina changes of mth-root metrics.

Investigate AR-Finsler metrics for local dual flatness and projective flatness.

Study on Gauduchon manifolds finds metrics for projectively flat bundles.

In this paper, we study a class of Finsler metrics composed by a Riemann metric $α=\sqrt{a_{ij}(x)y^i y^j}$ and a $1$-form $β=b_i(x)y^i$ called general ($α$, $β$)-metrics. We classify those projectively flat when $α$ is projectively flat. By solving the corresponding nonlinear PDEs, the metrics in this class are totall…

Study shows Ricci flat Calabi's metrics can't be projectively induced.

In Finsler geometry, there are infinitely many models of constant curvature. The Funk metrics, the Hilbert-Klein metrics and the Bryant metrics are projectively flat with non-zero constant curvature. A recent example constructed by the author is projectively flat with zero curvature. In this paper, we introduce a techn…

The class of spherically symmetric Finsler metrics is studied and locally dually flat and projectively flat spherically symmetric Finsler metrics is classified.

The paper extends Ricci curvature in Finsler geometry and finds conditions for specific metrics.

The paper studies projectively and dually flat Finsler spaces with Randers changes.

Stenzel's metrics can't be projectively induced in complex projective space.

Two conjectures on Ricci-flat metrics verified under specific conditions.

In this paper, a characteristic condition of the projectively flat Kropina metric is given. By it, we prove that a Kropina metric $F=α^2/β$ with constant curvature $K$ and $\|β\|_α=1$ is projectively flat if and only if $F$ is locally Minkowskian.

The well-known Funk metric F(x,y) is projectively flat with constant flag curvature K=-1/4 and the Hilbert metric H(x,y):=(F(x,y)+F(x,-y))/2 is projectively flat with constant curvature K=-1. These metrics are the special solutions to Hilbert's Fourth Problem. In this paper, we construct a non-trivial R-flat spray usin…

Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we classify a class of singular $(α,β)$-metrics which are locally projectively flat with constant flag curvature in dimension $n= 2$ and $n \ge 3$ respectively. Further, we determine t…

Research explores flat subspaces in complex projective manifolds using Okounkov bodies.

The paper studies scalar flat Kähler metrics on line bundles and proves their properties.

Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of singular Finsler metrics defined by a Riemann metric $α$ and 1-form $β$ and characterize those which are respectively Douglasian and locally projectively flat in di…

In this paper we study the flag curvature of a particular class of Finsler metrics called general $(α,β)$-metrics, which are defined by a Riemannian metric $α$ and a $1$-form $β$. The classification of such metrics with constant flag curvature are completely determined under some suitable conditions, which make them be…

In this paper, we study locally projectively flat Finsler metrics with constant flag curvature ${\bf K}$. We prove those are totally determined by their behaviors at the origin by solving some nonlinear PDEs. The classifications when ${\bf K}=0$, ${\bf K}=-1$ and ${\bf K} =1$ are given respectively in an algebraic way.…

Defines projective Ricci curvature and proves rigidity for sprays.

We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact …

Kähler cones over Sasakian manifolds are flat if projectively induced.

Study shows zero Pontrjagin classes for sprays.

In this paper, we study a class of two-dimensional Finsler metrics defined by a Riemannian metric $α$ and a 1-form $β$. We characterize those metrics which are Douglasian or locally projectively flat by some equations. In particular, it shows that the known fact that $β$ is always closed for those metrics in higher dim…

Study on spherical Finsler metrics with isotropic curvature rigidity.

A Riemannian metric is of constant curvature if and only if it is locally projectively flat. There are infinitely many locally projectively flat Finsler metrics of constant curvature, that are special solutions to the Hilbert's Fourth Problem. In this paper, we use the technique in the paper titled "Finsler metrics wit…

In this paper we study a class of Finsler metrics defined by a Riemannian metric and an 1-form. We classify those of projectively flat in dimension $n\geq3$ by a special class of deformations. The results show that the projective flatness of such kind of Finsler metrics always arises from that of some Riemannian metric…

The article studies Ricci-flat metrics on complex projective space.

Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of two-dimensional singular Finsler metrics defined by a Riemann metric $α$ and 1-form $β$, and we characterize those which are Douglasian or locally projectively flat…

Derives local forms for Bochner-flat (pseudo-)Kähler metrics.

The paper introduces a new spray deformation and finds new metrics.

The article constructs strong Carrollian geometries at infinity for Ricci flat Einstein manifolds.

In this work, the dual flatness, which is connected with Statistics and Information geometry, of general $(α,β)$-metrics (a new class of Finsler metrics) is studied. A nice characterization for such metrics to be dually flat under some suitable conditions is provided and all the solutions are completely determined. By …

Study classifies holonomy groups of projectively flat Randers surfaces.

Study spherically symmetric Finsler metrics with specific curvature properties.

For Finsler spaces (M,F) endowed with m-th root metrics, we provide necessary and sufficient conditions in which they are projectively flat, or projectively related to Berwald/Riemann spaces. We also give a specific characterization for m-th root metrics spaces of Landsberg and of Berwald type.

I give a theory of Moebius-flat hypersurfaces in n-dimensional projective space, analogous to that in conformal geometry. This unifies the classes of hypersurfaces with flat induced conformal structure (n > 3) and a classically studied class of surfaces (n = 3). I extend an example of Akivis-Konnov, and use polynomial …

The aim of this paper is to give a local description of affine surfaces, whose induced Blaschke structure is projectively flat. We show that such affine surfaces with constant Gauss affine curvature and indefinite induced Blaschke metric are described by soliton equations.

New Finsler metrics defined by Riemannian and 1-forms are studied.

An $(α,β)$-manifold $(M,F)$ is a Finsler manifold with the Finsler metric $F$ being defined by a Riemannian metric $α$ and $1$-form $β$ on the manifold $M$. In this paper, we classify $n$-dimensional $(α,β)$-manifolds (non-Randers type) which are positively complete and locally projectively flat. We show that the non-t…

We study the behaviour of families of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold when the Kahler classes degenerate to the boundary of the ample cone. We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ri…

We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact Kähler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they…

The expression (-1/u) times the Hessian of u transforms as a symmetric (0,2) tensor under projective coordinate transformations, so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold M, the section u can be regarded as a metric potential analogous to the local potential …

Uniform diameter bound for Ricci-flat Kahler metrics on projective manifolds.

Given a projective structure on a surface $N$, we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space $M$ of a certain rank $2$ affine bundle $M \to N$. The Einstein metric has anti-self-dual conformal curvature and admits …