Contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting the nearly trans-Sasakian structure

Abstract

In the present paper, we apply Hopf’s lemma to the contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting nearly trans-Sasakian structure and establish a characterization inequality for the existence of these types of warped products. This inequality generalizes the inequalities obtained in [M. Atceken, Bull. Iran. Math. Soc. – 2013. – 39, № 3. – P. 415 – 429; M. Atceken, Collect. Math. – 2011. – 62, № 1. – P. 17 – 26, and Sibel Sular, Cihan O¨ zgu¨r, Turkish J. Math. – 2012. – 36. – P. 485 – 497]. Moreover, we also compute another inequality for the squared norm of the second fundamental form in terms of warping functions. This inequality is a generalization of the inequalities acquired in [I. Mihai, Geom. Dedicata. – 2004. – 109. – P. 165 – 173 and K. Arslan, R. Ezentas, I. Mihai, C. Murathan, J. Korean Math. Soc. – 2005. – 42, № 5. – P. 1101 – 1110]. The inequalities proved in the paper either generalize or improve all inequalities available in the literature and related to the squared norm of the second fundamental form for contact CR-warped product of submanifolds of any almost contact metric manifold.