A Chen invariant on contact manifolds

Abstract

B.Y. Chen and the present author have defined some contact Riemannian invariants for almost contact metric manifolds analogous to Chen invariants. Sharp inequalities between these contact Riemannian invariants and the squared mean curvature for almost contact metric manifolds in a real space form were established. The equality case of these inequalities was investigated. On the other hand, we studied ideal C-totally real submanifolds in Sasakian space forms (see [?]). Ideal Legendrian submanifolds in a Sasakian space form are minimal. In the last section we improve Chen first inequality for such submanifolds and pointout that the equality case does not imply the minimality of the submanifold.