Complex Hypersurface Of Complex Finsler Manifolds

Let M be a complex manifold of complex dimension n and F be a strongly pseudoconvex complex Finsler metric on M [1]. The pair (M,F) is called a strongly pseudoconvex complex Finsler manifold. Let M be a complex sub-manifoldof M with complex dimension n-1, i.e., M is a complex hypersurface of M. Denote F the restriction of F to M, which is also called the induced complex Finsler metric by F on M. (M, F) is called a complex Finsler hypersurfaceof (M, F). Based on the work of [12],[13], the author studies in this paper some geometric properties of complex Finsler hypersurface of a Kaehler Finsler manifold. The main results of this paper are ( cf. Theorem 2.4, Theorem2.6- Theorem 2.8, Theorem 3.7):Theorem A Let (M,F) be a Kaehler Finsler manifold , (M, F) be a complex Finsler hypersurface of (M,F). Then the coefficients B_j;k of the second fundamental form B(·,·) for (M, F) are given byTheorem B Let (M,F) be a Kaehler Finsler manifold , (M, F) be a complex Finsler hypersurface of (M, F) which is not totally geodesic. Then the following equalityholds if and only ifTheorem C Let (M, F) be a Kaehler Finsler manifold , (M, F) be a complex Finsler hypersurface of (M, F) which is not totally geodesic. Let D be the complex Rund connection associated to (M,F). Then M_jⁱ = 0 if and only if the induced complex linear connection (?) by D on (M, F) coincides with the intrinsic complex Rund connection (?) associated to (M, F). Theorem D Let (M, F) be a complex Berwald manifold , (M, F) be a complex Finsler hypersurface of (M, F). If (M, F) is totally geodesic andThen (M,F) is a complex Berwald submanifold.Theorem E Let (M, F) be a strongly pseudoconvex locally complex Minkowski manifold endowed with complex Rund connection. Let (M, F) be a complex Finsler hypersurface of (M, F) which satisfies(i) M_k^l= M_?^l = M_i = 0 on (?) (?) i, l, k∈{1,…, n -1}.(ii) The matrix (?) is nonsingular at every point of M.Then F is a Hermitian metric on M.