SOME WARPED PRODUCT SUBMANIFOLDS OF A KENMOTSU MANIFOLD

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 51, Issue 3, 2014, pp.863-881
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2014.51.3.863

Title & Authors

SOME WARPED PRODUCT SUBMANIFOLDS OF A KENMOTSU MANIFOLD

Khan, Viqar Azam; Shuaib, Mohammad;

Khan, Viqar Azam; Shuaib, Mohammad;

Abstract

Many differential geometric properties of a submanifold of a Kaehler manifold are conceived via canonical structure tensors T and F on the submanifold. For instance, a CR-submanifold of a Kaehler manifold is a CR-product if and only if T is parallel on the submanifold (c.f. [2]). Warped product submanifolds are generalized version of CR-product submanifolds. Therefore, it is natural to see how the non-triviality of the covariant derivatives of T and F gives rise to warped product submanifolds. In the present article, we have worked out characterizations in terms of T and F under which a contact CR- submanifold of a Kenmotsu manifold reduces to a warped product submanifold.

Keywords

CR-submanifold;warped product;Kenmotsu manifold;

Language

English

References

1.

R. L. Bishop and B. O'Neil, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-9.

2.

B. Y. Chen, CR-submanifold of Kaehler manifolds. I, J. Differential Geom. 16 (1981), no. 2, 305-323.

4.

B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds, Montash. Math. 133 (2001), no. 3, 177-195.

5.

I. Hasegawa, I. Mihai, Contact CR-warped product submanifolds in Sasakian manifolds, Geom. Dedicata 102 (2003), 143-150.

6.

S. Hiepko, Eine innere Kennzeichnung der verzerrten Produkte, Math. Ann. 241 (1979), no. 3, 209-215.

7.

S. T. Hong, Warped product and black holes, Nuovo Cimento Soc. Ital. Fis. B 120 (2005), no. 10-11, 1221-1234.

9.

V. A. Khan, K. A. Khan, and Siraj-Uddin, CR-Warped product submanifolds in a Kaehler Manifold, Southeast Asian Bull. Math. 33 (2009), no. 5, 865-874.

10.

M. Kobayashi, Semi-invariant submanifolds of a certain class of almost contact manifolds, Tensor (N.S.) 43 (1986), no. 1, 28-36.

11.

M. I. Munteanu, Warped product contact CR-submanifold of Sasakian space form, Publ. Math. Debreen 66 (2005), no. 1-2, 75-120.

12.

M. I. Munteanu, A note on doubly warped product contact CR-submanifold in trans-Sasakian manifolds, Acta Math. Hungar. 16 (2007), no. 1-2, 121-126.

13.

J. A. Oubina, New class of almost contact metric structures, Publ. Math. Debreen 32 (1985), no. 3-4, 187-193.

14.

S. Tanno, The automorphism groups of almost contact Riemannian manfolds, Tohoku Math. J. 21 (1969), 21-38.

15.

K. Yano and M. Kon, CR-Submanifolds of Kaehlerian and Sasakian Manifolds, Birkhauser Verlag, Boston, 1983.