Hermitian-Poisson Metrics on Flat Bundles over Complete Hermitian Manifolds

In this paper, the author solves the Dirichlet problem for Hermitian-Poisson metric equation √?1Λ_ωG_H = λId and proves the existence of Hermitian-Poisson metrics on flat bundles over a class of complete Hermitian manifolds. When λ = 0, the HermitianPoisson metric is a Hermitian harmonic metric.     

Geometric properties of almost contact metric 3-submersions

Summary In this paper, we discuss some geometric properties of three types of Riemannian submersions whose total space is an almost contact metric manifold with 3-structure. The study is focused on the transference of structures.     

On Compact Hermitian Manifolds with Flat Gauduchon Connections

Given a Hermitian manifold(M~n, g), the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call▽~s=(1-s/2)▽~c+s/2▽~b the s-Gauduchon connection of M, where ▽~c and ▽~b are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat s-Gauduchon connection. This is related to a question asked by Yau. The cases with s = 0(a flat Chern connection) or s = 2(a flat Bismut connection) are classified respectively by Boothby in the1950 s or by the authors in a recent joint work with Q. Wang. In this article, we observe that if either s ≥ 4 + 2×3~(1/2) ≈ 7.46 or s ≤ 4-2×3~(1/2) ≈ 0.54 and s ≠ 0, then g is K?hler. We also show that, when n = 2,g is always K?hler unless s = 2. Therefore non-K?hler compact Gauduchon flat surfaces are exactly isosceles Hopf surfaces.     

On Einstein extensions of nilpotent metric Lie algebras

The main result of the article is as follows: If a nilpotent noncommutative metric Lie algebra (n, Q) is such that the operator Id ? trace(Ric) / trace(Ric²) Ric is positive definite then every Einstein solvable extension of (n, Q) is standard. We deduce several consequences of this assertion. In particular, we prove that all Einstein solvmanifolds of dimension at most 7 are standard.     

On contact metric statistical manifolds

A notion of almost contact metric statistical structure is introduced and thereby contact metric and K-contact statistical structures are defined. Furthermore a necessary and sufficient condition for a contact metric statistical manifold to admit K-contact statistical structure is given. Finally, the condition for an odd-dimensional statistical manifold to have K-contact statistical structure is expressed.     

Noncompact quasi‐Einstein manifolds conformal to a Euclidean space

The goal of this article is to investigate nontrivial m‐quasi‐Einstein manifolds globally conformal to an n‐dimensional Euclidean space. By considering such manifolds, whose conformal factors and potential functions are invariant under the action of an $(n?1)$‐dimensional translation group, we provide a complete classification when $\lambda =0$ and $m\ge 1$ or $m=2?n$.     

On Einstein Matsumoto metrics

We study a special class of Finsler metrics,namely,Matsumoto metrics F=α2α-β,whereαis a Riemannian metric andβis a 1-form on a manifold M.We prove that F is a(weak)Einstein metric if and only ifαis Ricci flat andβis a parallel 1-form with respect toα.In this case,F is Ricci flat and Berwaldian.As an application,we determine the local structure and prove the 3-dimensional rigidity theorem for a(weak)Einstein Matsumoto metric.     

Some remarks on Kong-Liu’s conjecture on Lorentzian metrics

For a(1+3)-dimensional Lorentzian manifold(M,g),the general form of solutions of the Einstein field equations takes that of type I,II,or III.For type I,there is a known result in Gu(2007).In this paper,we try to find the necessary and sufficient conditions for the Lorentzian metric to take the form of types II and III,and we show how to construct the new coordinate system.     

On Einstein Hermitian Surfaces

We show that every compact Einstein Hermitian surface with constant conformal scalar curvature is a Kahler surface and that, in contrast to the compact case, there exits a noncompact Einstein Hermitian and non-Kahler surface with constant conformal scalar curvature.