The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair consisting of a left‐invariant Riemannian metric g and a positive constant c such that , where is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that is solvable for some left‐invariant Riemannian metric g.     

Parallel and totally geodesic hypersurfaces of non-reductive homogeneous four-manifolds

We classify totally geodesic and parallel hypersurfaces of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds.     

一类对偶平坦的球对称的芬斯勒度量

本文研究了对偶平坦的芬斯勒度量的构造问题.通过分析球对称的对偶平坦的芬斯勒度量的方程的解,我们构造了一类新的对偶平坦的芬斯勒度量,并得到了球对称的芬斯勒度量成为对偶平坦的充分必要条件.     

A class of Finsler metrics of scalar flag curvature

It is known that every locally projectively flat Finsler metric is of scalar flag curvature. Conversely, it may not be true. In this paper, for a certain class of Finsler metrics, we prove that it is locally projectively flat if and only if it is of scalar flag curvature. Moreover, we establish a class of new non-trivial examples.     

Bochner-Kodaira Techniques on K¨ahler Finsler Manifolds

A horizontal Hodge Laplacian operator $\square_{\mathcal {H}}$ is defined for Hermitian holomorphic vector bundles over PTM on K¨ahler Finsler manifold, and the expression of $\square_{\mathcal {H}}$ is obtained explicitly in terms of horizontal covariant derivatives of the Chern-Finsler connection. The vanishing theorem is obtained by using the $\partial_{\mathcal {H}}\ov{\partial}_{\mathcal {H}}$-method on K¨ahler Finsler manifolds.