An inequality for contact CR-submanifolds in Sasakian space forms

Abstract

The theory of $$\delta $$ -invariants was initiated by Chen (Arch Math 60:568–578, 1993) in order to find new necessary conditions for a Riemannian manifold to admit a minimal isometric immersion in a Euclidean space. Chen (Int J Math 23(3):1250045, 2012) defined a CR $$\delta $$ -invariant $$\delta (D)$$ for anti-holomorphic submanifolds in complex space forms. Afterwards, Al-Solamy et al. (Taiwan J Math 18:199–217, 2014) established an optimal inequality for this invariant for anti-holomorphic submanifolds in complex space forms. In this article, we prove an optimal inequality for the contact CR $$\delta $$ -invariant on contact CR-submanifolds in Sasakian space forms. An example for the equality case is given.