On totally real statistical submanifolds

Abstract

In the present paper, first we prove some results by using fundamental properties of totally real statistical submanifolds immersed into holomorphic statistical manifolds. Further, we obtain the generalized Wintgen inequality for Lagrangian statistical submanifolds of holomorphic statistical manifolds with constant holomorphic sectional curvature c. The paper finishes with some geometric consequences of obtained results.

Optimizations on Statistical Hypersurfaces with Casorati Curvatures

“…for any X, Y ∈ Γ(T M). Then, by simple computation, we see that K 1 satisfies three conditions of Lemma 2.1, and hence a holomorphic statistical manifold (M, ∇ := 26]). For a Kaehler manifold (M, g, J), we take a vector field U ∈ Γ(T M) and set K 2 as follows: 2) ) satisfies three conditions of Lemma 2.1 as in Example 2.2, and hence (M, ∇ := ∇ g + K 2 , g, J) becomes a holomorphic statistical manifold.…”

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

“…Especially, we give all the Sasakian statistical structures on the usual Sasakian manifold R 3 in terms of three independent functions(see Proposition 5.1). Moreover, we find out all the holomorphic statistical structures of constant holomorphic curvature 0 on a Kähler manifold due to A. N. Siddiqui and M. H. Shahid [17] It can be easily proved that (∇ X g)(Y, Z) = (∇ Y g)(X, Z) and K(e i , e j ) = K(e j , e i ), g(K(e i , e j ), e k ) = g(K(e i , e k ), e j ), K(e i , φe j ) + φK(e i , e j ) = 0, or equivalently, the coefficients {K l ij } satisfy:…”

“…Example 5.2 and Example 5.3).Especially, we give all the Sasakian statistical structures on the usual Sasakian manifold R 3 in terms of three independent functions(see Proposition 5.1). Moreover, we find out all the holomorphic statistical structures of constant holomorphic curvature 0 on a Kähler manifold due to A. N. Siddiqui and M. H. Shahid[17](see Proposition 5.2). Consider a 3-dimensional manifold M = {(x, y, z) ∈ R 3 |y = 0}, where (x, y, z) are the standard coordinate system in R 3 .…”

“…Lemma 5.1. [17] Consider a 2−dimensional manifold M = {(x, y) ∈ R 2 |x > 0, y > 0}, where (x, y) is the standard coordinate system in R 2 . The Riemannian metric g on M is given by g = x{(dx) 2 + (dy) 2 } and the complex structure J is defined by…”

Recent developments in Wintgen inequality and Wintgen ideal submanifolds

“…Let M m be a Kaehler manifold of complex dimension m equipped with an almost complex structure J and a Kaehlerian metric g. A quadruple (M m , ∇, g, J) is called a holomorphic statistical manifold (see, e.g., [125,127]) if Then…”