IJPAM: Volume 51, No. 2 (2009)

GENERAL SUBMANIFOLDS OF A KAEHLER MANIFOLD

Ameera A. Eshan$^1$, Falleh R. Al-Solamy$^2$
$^1$Department of Mathematics
Taif University
P.O. Box 20314, Makkah, 21599, KINGDOM OF SAUDI ARABIA
e-mail: [email protected]
$^2$Department of Mathematics
University of Tabuk
P.O. Box 80015, Jeddah, 21589, KINGDOM OF SAUDI ARABIA
e-mail: [email protected]


Abstract.In this paper we initiate the study of most general class of submanifolds of a Kaehler manifold which includes all existing classes of submanifolds (complex submanifolds, totally real submanifolds, CR-submanifolds, slant submanifolds). Such a submanifold $M$ of a Kaehler manifold $\overline{M}$ has naturally defined operators $\phi$, $F$, $\psi$ and $G$ (see Section 3 for the definition). First we study the basic properties of these operators for a general submanifold and latter we characterize submanifolds with parallel $\phi$ and show that essentially such submanifolds of a Kaehler manifold are CR-submanifold. We also give examples of submanifolds of a Kaehler manifold which have parallel $\phi$ and a submanifold on which the operator $\phi$ is not parallel.

Received: January 26, 2009

AMS Subject Classification: 53C20, 58C45

Key Words and Phrases: complex submanifolds, totally real submanifolds, CR-submanifolds, slant submanifolds, Kaehler manifolds

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 51
Issue: 2