复Finsler流形上的Koppelman-Leray公式

利用不变积分核(Berndtsson核)、复Finsler度量和联系于Chern-Finsler联络的非线性联络来研究复Finsler流形上的积分表示理论,得到了Koppelman公式和Koppelman-Leray公式,并给出了∂-方程的解.     

复Finsler流形上的Koppelman-Leray-Norguet公式

利用不变积分核(Berndtsson核),复Finsler度量和联系于Chern-Finsler联络的非线性联络,研究复Finsler流形上具有逐块光滑C~((1))边界的有界域上(p,q)型微分形式的积分表示,得到了(p,q)型微分形式的Koppelman-Leray-Norguet公式和■-方程的解．作为应用,利用复Finsler度量和联系于Chern-Finsler联络的非线性联络,给出了Stein流形上具有逐块光滑C~((1))边界的有界域上(p,q)型微分形式的Koppelman- Leray-Norguet公式以及■-方程的解,并且得到了Stein流形上实非退化强拟凸多面体上(p,q)型微分形式的积分表示式和■-方程的解．     

复Finsler流形上(p,q)型微分形式的积分公式

利用Demailly和Laurent-Thiebaut不变积分核,复Finsler度量和联系于Chern-Finsler联络的非线性联络来研究复Finsler流形上的积分表示理论,得到了Koppelman和Koppelman-Leray公式, 并给出了 ∂-方程的解.     

关于复Finsler流形的一个注记

设(M,G)为n维复Finsler流形,TM为M的全纯切丛,得到了TM上的Hermite度量hTM=G(-ij)(z,v)dzi(×)d(-z)j+G(-ij)(z,v)δvi(×)δ(-v)j为K(a)hler度量的充要条件是M为全纯曲率为0的Kahler流形,其中G(-ij)=(а)2G/(а)vi(а)(-v)j,1≤i,j≤n.推广了Cao-Wong的某些结果.     

具有（α，β）度量的复Finsler流形

设M是复流形,具有复（α,β）度量F=αφ（｜β｜/α）,其中α为M上的Hermite度量,β为M上的（1,0）形式。本文得到与F相联系的复非线性联络系数Гiμ^i的表达式,且证明了：若β为M上的全纯（1,0）形式,并且关于α的Hermite联络γij^k（z）平行,则F是M上的复Berwald度量;若α是M上的Kaihler度量,则F是M上的强Kahler Finsler度量．     

复Finsler流形上的Hodge-Laplace算子

本文定义了强拟凸复Finsler流形上的Hodge-Laplace算子,并给出其水平部分的局部坐标表示．     

复Finsler子流形的全纯曲率

设M为n维复流形,F为M上的强拟凸的复Finsler度量,M是M的m维复子流形,F是F在M上诱导的复Finsler度量,D为（M,F）上的复Rund联络．本文证明了（1）（M,F）上的诱导复线性联络△↓的全纯曲率与（M,F）上的复Rund联络△↓^＊的全纯曲率相同;（2）联络△↓^＊的全纯曲率不超过联络D的全纯曲率;（3）（M,F）是（M,F）的全测地复Finsler子流形的充分必要条件是（M,F）的第2基本形式B（．,．）的适当形式的缩并为零,即B（x,l）=0．本文的证明主要利用复Finsler子流形（M,F）的Gauss,Codazzi和Ricci方程．     

Finsler流形上的Laplace算子

该文对Finsler流形上的微分式定义了整体内积，进而引入δ算子和Laplace算子。该文还给出了δ算子的局部坐标表达式并且证明了Laplace算子可以看成是Riemann流形上Laplace算子在Finsler流形上的扩张。     

Stein流形上全纯函数积分公式的拓广

得到了Stein流形上一种全纯函数的积分式，这种公式的特点含有可供选择的参数m≥2的整数，当m=2时即为Stein流形上全纯函数的方B-M公式．     

Laplacian on Complex Finsler Manifolds

In this paper, the Laplacian on the holomorphic tangent bundle T1,0M of a complex manifold M endowed with a strongly pseudoconvex complex Finsler metric is defined and its explicit expression is obtained by using the Chern Finsler connection associated with (M,F). Utilizing the initiated “Bochner technique”, a vanishing theorem for vector fields on the holomorphic tangent bundle T1,0M is obtained.     

Holomorphic curvature of complex Finsler submanifolds

Let M be a complex n-dimensional manifold endowed with a strongly pseudoconvex complex Finsler metric F. Let M be a complex m-dimensional submanifold of M, which is endowed with the induced complex Finsler metric F. Let D be the complex Rund connection associated with (M, F). We prove that (a) the holomorphic curvature of the induced complex linear connection  on (M, F) and the holomorphic curvature of the intrinsic complex Rund connection ～* on (M, F) coincide; (b) the holomorphic curvature of ～* does not exceed the holomorphic curvature of D; (c) (M, F) is totally geodesic in (M, F) if and only if a suitable contraction of the second fundamental form B(·, ·) of (M, F) vanishes, i.e., B(χ, ι) = 0. Our proofs are mainly based on the Gauss, Codazzi and Ricci equations for (M, F).     

乘积复流形上的Szabó度量

设(M_1,α),(M_2,β)均为Hermitian流形,本文证明了积流形M_1×M_2上的复Szabó度量F_ε是Berwald度量,且当α,β为K(?)hler度量时,F_ε是强Kahler-Finsler度量,此外本文还给出了F_ε的全纯曲率的显式表达式．     

乘积复流形上的Szabó度量

设(M_1,α),(M_2,β)均为Hermitian流形,本文证明了积流形M_1×M_2上的复Szabó度量F_ε是Berwald度量,且当α,β为K(?)hler度量时,F_ε是强Kahler-Finsler度量,此外本文还给出了F_ε的全纯曲率的显式表达式．     

On Unitary Invariant Strongly Pseudoconvex Complex Landsberg Metrics

In this paper,we prove that a unitary invariant strongly pseudoconvex complex Finsler metric is a complex Landsberg metric if and if only if it comes from a unitary invariant Hermitian metric. This implies that there does not exist unitary invariant complex Landsberg metric unless it comes from a unitary invariant Hermitian metric.     

K(a)hler Finsler Metrics Are Actually Strongly K(a)hler

In this paper,the K(a)hler conditions of the Chern-Finsler connection in complex Finsler geometry are studied,and it is proved that K(a)hler Finsler metrics are actually strongly K(a)hler.     

On dually flat Finsler metrics

In this paper, we study the locally dually flat Finsler metrics which arise from information geometry. An equivalent condition of locally dually flat Finsler metrics is given. We find a new method to construct locally dually flat Finsler metrics by using a projectively flat Finsler metric under the condition that the projective factor is also a Finsler metric. Finally, we find that many known Finsler metrics are locally dually flat Finsler metrics determined by some projectively flat Finsler metrics.     

Rigidity of Noncompact Finsler Manifolds

For a Euclidean space or a Minkowski space, we change the metric in a compact subset and show that the resulting Finsler manifold is isometric to the original standard space under certain conditions. We assume that the mean tangent curvature vanishes and the metric satisfies some curvature conditions or have no conjugate points.