Totally Real Statistical Submanifolds

Abstract

We prove that a semi-parallel totally real statistical submanifold with some natural conditions is totally geodesic if it is of non zero constant curvature, which is corresponding to the Kassabov theorem in the submanifold theory of Kähler manifolds. Moreover, we construct four dimensional holomorphic statistical manifolds using g-natural metrics (cf.[1]).

Inequalities for the Casorati Curvature of Statistical Manifolds in Holomorphic Statistical Manifolds of Constant Holomorphic Curvature

“…Therefore, the function v becomes v(x 1 , x 2 , y 1 , y 2 ) = 1. Then, for the metric G and the complex structure J, there exists a tensor field K such that (R 4 ,∇ := ∇ G + K,g := G, J) is a special Kähler manifold [46]. Notice that a holomorphic statistical structure of holomorphic curvature 0 is nothing but a special Kähler manifold [43].…”

“…where u y 1 := ∂u ∂y 1 and u y 2 := ∂u ∂y 2 . If K performs the conditions in Equations (10)- (12) and also the conditions in Equations (29), (30), then we get (R 4 ,∇ := ∇ G + K,g := G, J) a special Kähler manifold [46] with K constructed as follows: Then,M = (R 4 ,∇ := ∇ G + K, G, J) is a holomorphic statistical manifold of holomorphic curvature 0. Next, let M be any m-dimensional submanifold (m < 4) ofM.…”

On CR-statistical submanifolds of holomorphic statistical manifolds

“…Boyom et al [13] studied classification of totally umbilical CR-statistical submanifolds in holomorphic statistical manifolds with constant holomorphic curvature. Also, Milijević [12] and Siddiqui et al [20] studied totally real statistical submanifolds in holomorphic statistical manifolds, independently.…”

Some Results on Statistical Hypersurfaces of Sasakian Statistical Manifolds and Holomorphic Statistical Manifolds

“…Also, some other results in Riemannian geometry can be generalized to the geometry of statistical manifolds. For example, in 2015, M. Milijević [14] generalized a classical result [11] on totally real submanifolds of complex space forms to totally real statistical submanifolds of holomorphic statistical manifolds. Recently, M. Milijević [15] proved the non-existence of CR submanifolds of maximal CR dimension with umbilical shape operators in holomorphic statistical manifolds of nonzero constant holomorphic sectional curvature.…”