Chen-like Inequalities for Submanifolds in Kähler Manifolds Admitting Semi-Symmetric Non-Metric Connections

Abstract

The geometry of submanifolds in Kähler manifolds is an important research topic. In the present paper, we study submanifolds in complex space forms admitting a semi-symmetric non-metric connection. We prove the Chen–Ricci inequality, Chen basic inequality, and a generalized Euler inequality for such submanifolds. These inequalities provide estimations of the mean curvature (the main extrinsic invariants) in terms of intrinsic invariants: Ricci curvature, the Chen invariant, and scalar curvature. In the proofs, we use the sectional curvature of a semi-symmetric, non-metric connection recently defined by A. Mihai and the first author, as well as its properties.

Keywords

• Kähler manifold
• complex space form
• submanifolds
• mean curvature
• Ricci curvature
• semi-symmetric connection