On U(n)-invariant strongly convex complex Finsler metrics

In this paper,we obtain a necessary and sufficient condition for a U(n)-invariant complex Finsler metric F on domains in C^(n) to be strongly convex,which also makes it possible to investigate the relationship between real and complex Finsler geometries via concrete and computable examples.We prove a rigid theorem which states that a U(n)-invariant strongly convex complex Finsler metric F is a real Berwald metric if and only if F comes from a U(n)-invariant Hermitian metric.We give a characterization of U(n)-invariant weakly complex Berwald metrics with vanishing holomorphic sectional curvature and obtain an explicit formula for holomorphic curvature of the U(n)-invariant strongly pseudoconvex complex Finsler metric.Finally,we prove that the real geodesics of some U(n)-invariant complex Finsler metric restricted on the unit sphere S^(2n-1)■C^(n) share a specific property as that of the complex Wrona metric on C^(n).

ON DOUBLY WARPED PRODUCT OF COMPLEX FINSLER MANIFOLDS

Let （M1, F1） and （M2, F2） be two strongly pseudoconvex complex Finsler man- ifolds. The doubly wraped product complex Finsler manifold （f2 M1 x h M2, F） of （M1, F1） and （M2, F2） is the product manifold M1 x M2 endowed with the warped product complex 2 2 Finsler metric F2 = f2F1 ＋ fl F2, where fl and f2 are positive smooth functions on M1 and M2, respectively. In this paper, the most often used complex Finsler connections, holomorphic curvature, Ricci scalar curvature, and real geodesics of the DWP-complex Finsler manifold are derived in terms of the corresponding objects of its components. Necessary and sufficient conditions for the DWP-complex Finsler manifold to be K/ihler Finsler （resp., weakly K/ihler Finsler, complex Berwald, weakly complex Berwald, complex Landsberg） manifold are ob- tained, respectively. It is proved that if （M1, F1） and （M2,F2） are projectively flat, then the DWP-complex Finsler manifold is projectively flat if and only if fl and f2 are positive constants.

Locally Conformal Pseudo-Kahler Finsler Manifolds

In this paper,we give a necessary and sucient condition for a strongly pseudoconvex complex Finsler metric to be locally conformal pseudo-Kahler Finsler.As an application,we nd any complete strongly convex and locally conformal pseudo-Kahler Finsler manifold,which is simply connected or whose fundamental group contains elements of nite order only,can be given a Kahler metric.

On Unitary Invariant Weakly Complex Berwald Metrics with Vanishing Holomorphic Curvature and Closed Geodesics

In this paper, the authors construct a class of unitary invariant strongly pseudoconvex complex Finsler metrics which are of the form F =√[ rf(s- t)[, where r = ||v||~ 2, s =| z,v |~2/r, t =|| z||~ 2, f(w) is a real-valued smooth positive function of w ∈ R,and z is in a unitary invariant domain M C^n. Complex Finsler metrics of this form are unitary invariant. We prove that F is a class of weakly complex Berwald metrics whose holomorphic curvature and Ricci scalar curvature vanish identically and are independent of the choice of the function f. Under initial value conditions on f and its derivative f, we prove that all the real geodesics of F =√[rf(s- t)] on every Euclidean sphere S^(2n-1) M are great circles.

Khler Finsler Metrics Are Actually Strongly Khler

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

Laplacian on Complex Finsler Manifolds

In this paper, the Laplacian on the holomorphic tangent bundle T 1,0 M 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 T 1,0 M is obtained.

Strongly convex weakly complex Berwald metrics and real Landsberg metrics

Under the assumption that' is a strongly convex weakly Khler Finsler metric on a complex manifold M, we prove that F is a weakly complex Berwald metric if and only if F is a real Landsberg metric.This result together with Zhong(2011) implies that among the strongly convex weakly Kahler Finsler metrics there does not exist unicorn metric in the sense of Bao(2007). We also give an explicit example of strongly convex Kahler Finsler metric which is simultaneously a complex Berwald metric, a complex Landsberg metric,a real Berwald metric, and a real Landsberg metric.