Application of Hopf's lemma on contact CR-warped product submanifolds of a nearly Kenmotsu manifold

Document Type : Research Paper


1 ‎Department of Mathematics, ‎University of Tabuk‎, ‎Kingdom of Saudi Arabia.

2 ‎Department of Mathematics, ‎King Abdulaziz University, ‎P.O‎. ‎Box 80015‎, ‎Jeddah 21589‎, ‎Kingdom of Saudi Arabia.


In this paper we consider contact CR-warped product submanifolds of the type $M = N_T\times_f N_\perp$, of a nearly Kenmotsu generalized Sasakian space form $\bar M(f_1‎, ‎f_2‎, ‎f_3)$ and by use of Hopf's Lemma we show that $M$ is simply contact CR-product under certain condition‎. ‎Finally‎, ‎we establish a sharp inequality for squared norm of the second fundamental form and equality case is discussed‎. ‎The results in this paper generalize existing results in the literature.


Main Subjects

P. Alegre, D. E. Blair and A. Carriazo, Generalized Sasakian space forms, Israel J. Math. 141 (2004) 157--183.
K. Arslan, R. Ezentas, I. Mihai and C. Murathan, Contact CR-warped product submanifolds in Kenmotsu space forms, J. Korean Math. Soc. 42 (2005), no. 5, 1101--1110.
F. R. Al-Solamy and M. A. Khan, Semi-slant warped product submanifolds of Kenmotsu manifolds, Math. Probl. Eng. 2012 (2012), Article ID 708191, 10 pages.
M. Atceken, Contact CR-warped product submanifolds in Kenmotsu space forms, Bull. Iranian Math. Soc. 39 (2013), no. 3, 415--429.
R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969) 1--49.
D. E. Blair, Contact manifolds in Riemannian Geometry, Lecture Notes in Math. 509, Springer-Verlag, Berlin, 1976.
D. E. Blair and D. K. Showers, Almost contact manifolds with Killing structure tensors II, J. Differential Geom. 9 (1974) 577--582.
B. Y. Chen, CR-submanifolds of a Kaehler manifold I, J. Differential Geom. 16 (1981), no. 2, 305--322.
B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds I, Monatsh. Math. 133 (2001), no. 3, 177--195.
B. Y. Chen, Pseudo-Riemannian Geometry, invariants and Applications, World Sci. Publ. Singapore, 2011.
B. Y. Chen, A survey on geometry of warped product submanifolds, ArXiv:1307.0236 [math. DG] (2013).
I. Hasegawa and I. Mihai, Contact CR-warped product submanifolds in Sasakian manifolds, Geom. Dedicata 102 (2003) 143--150.
K. Kenmotsu, A class of almost contact Riemannian manifolds, T^ohoku Math. J. 24 (1972) 93--103.
M. A. Khan, S. Uddin and R. Sachdevadeva, Semi-invariant warped product submanifolds of cosymplectic manifolds, J. Inequal. Appl. 2012 (2012), no. 19, 12 pages.
G. D. Ludden, Submanifolds of cosymplectic manifolds, J. Differential Geom. 4 (1970) 237--244.
I. Mihai, Contact CR-warped product submanifolds in Sasakain space forms, Geom. Dedicata 109 (2004) 165--173.
A. Mustafa, S. Uddin, V. A. Khan and B. R. Wong, Contact CR-warped product submanifolds of nearly trans-Sasakian manifolds, Taiwanese J. Math. 17 (2013), no. 4, 1473--1486.