C-totally real warped product submanifolds

We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequalities involving the Laplacian of the warping function and the squared mean curvature of $C$-totally real warped product submanifolds of $(\kappa ,\mu )$-space forms, Sasakian space forms and non-Sasakian $(\kappa ,\mu)$-manifolds. Then we obtain obstructions to the existence of minimal isometric immersions of $C$-totally real warped product submanifolds in $(\kappa ,\mu)$-space forms, non-Sasakian $(\kappa ,\mu)$-manifolds and Sasakian space forms. In the last, we obtain an example of a warped product $C$-totally real submanifold of a non-Sasakian $(\kappa ,\mu)$-manifold, which satisfies the equality case of the basic inequality.