Submanifold Theory in Holomorphic Statistical Manifolds

Curvature bounded conjugate symmetric statistical structures with complete metric

“…Moreover, the assumption about the constant curvature is replaced by the assumption that the curvature satisfies some inequalities. Since the notion of the sectional ∇-curvature is relatively new, see [6], [1], the theorems proved in this paper show that the notion is meaningful.…”

“…It is clear that a statistical structure can be defined equivalently as a pair (g, K), where K is a symmetric tensor field of type (1,2) which is also symmetric relative to g, or as a pair (g, A), where A is a symmetric cubic form.…”

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

“…Let S be the statistical curvature tensor field of a statistical manifold (M, g, ∇), where S ∈ Γ(TM (1,3) ) is defined by [40]…”

“…If π = span R {u 1 , u 2 } is a 2-dimensional subspace of T p M, for p ∈ M, then the sectional curvature of M is defined by [40]:…”

“…is called totally geodesic with respect to the connection∇ if the second fundamental form h of M for∇ vanishes identically [40].…”