Curvature Invariants on Partially Totally Real Submanifolds in Complex Space Forms

Kähler manifolds are the most studied complex manifolds. A Kähler manifold with constant holomorphic sectional curvature is said to be a complex space form. There are special classes of submanifolds in Kähler manifolds. Recently, Poyraz et al. defined partially totally real submanifolds in complex space forms. In the present paper, we study curvature invariants adapted to partially totally real submanifolds. First, we define a Riemannian invariant, denoted by δPTR(D) extending the definition of Chen’s CRδ-invariant and estimate it in terms of the squared mean curvature. Furthermore, we obtain Chen–Ricci inequalities that distinguish between tangent directions in the two distributions. Finally, we prove a Chen first inequality for totally real two-plane sections.