Abstract
In this paper, we prove some inequalities in terms of the normalized δ -Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of statistical submanifolds in holomorphic statistical manifolds with constant holomorphic sectional curvature. Moreover, we study the equality cases of such inequalities. An example on these submanifolds is presented.
Highlights
The problem of discovering simple relationships between the main intrinsic invariants and the main extrinsic invariants of submanifolds is a basic problem in submanifold theory [1]
Chen demonstrated in [3] an optimal inequality for a submanifold on a real space form between the intrinsically defined δ-curvature and the extrinsically defined squared mean curvature
This approach initiated a new line of research and was extended to various types of submanifolds in several types of ambient spaces, e.g., submanifolds in complex space forms of constant holomorphic sectional curvature
Summary
The problem of discovering simple relationships between the main intrinsic invariants and the main extrinsic invariants of submanifolds is a basic problem in submanifold theory [1]. Vîlcu established an optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature [22]. Denote by δC∗ (r; m − 1) and δ∗ C (r; m − 1) the dual generalized normalized δ∗ -Casorati curvatures of the submanifold M, defined as follows: δC∗ (r; m − 1)| p = r C ∗ | p + a(r ) inf{C ∗ ( L) | L a hyperplane of Tp M }, if 0 < r < m(m − 1), and δC∗ (r; m − 1)| p = r C ∗ | p + a(r ) sup{C ∗ ( L) | L a hyperplane of Tp M }, if r > m(m − 1), for a(r ) set above. M, the critical points of f | M are global optimal solutions of the Problem (14)
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