Research On Some Problems Of Sasakian Statistical Manifolds And Their Submanifolds

An important problem in differential geometry is to study the properties of various geometric spaces and establish the corresponding submanifold theory.As known that,a Sasakian statistical manifold can be viewed as a generalization of the classical Sasakian manifold.The main purpose of this paper is to study some properties of the Sasakian statistical manifolds and their submanifolds,specially speaking:(1)Several examples of the Sasakian statistical structure based on Sasakian manifolds are given.(2)We prove that if the Sasakian statistical structure based on a Sasakian space form of constant φ-sectional curvature c is of constant φ-sectional curvature ?c with respect to the statistical curvature tensor field,then ?c c,and the Sasakian statistical structure which satisfies the equality is also determined.(3)For the Sasakian statistical manifold of constant φ-sectional curvature,we prove that if there exists the statistical hypersurface or the contact CR-statistical submanifold of maximal contact CR-dimension satisfying some umbilical conditions,then the constant φ-sectional curvature of the ambient space equals to 1.(4)We study two kinds of statistical immersions with codimension 1 between a Sasakian statistical manifold of constant φ-curvature and a statistical manifold of constant curvature,and prove that in both cases the φ-curvature equals to 1.(5)Further,we consider two kinds of statistical immersions with codimension 1 between a Sasakian statistical manifold of constant φ-curvature and a holomorphic statistical manifold of constant holomorphic curvature,and obtain the results similar to(4).(6)We study C-totally real statistical submanifolds and CR-totally real statistical submanifolds of a Sasakian statistical manifold.We obtain some conditions for C-totally real statistical submanifolds to be geodesic,and prove that under the similar conditions CR-totally real statistical submanifolds must be minimal but not totally umbilical.