TAILIEUCHUNG - On invariant submanifolds of Riemannian warped product manifold

In this paper, we generalize the geometry of the invariant submanifolds of Riemannian product manifold to the geometry of the invariant submanifolds of Riemannian warped product manifold. | Turk J Math 27 (2003) , 407 – 423. ¨ ITAK ˙ c TUB On Invariant Submanifolds of Riemannian Warped Product Manifold M. At¸ceken, B. S ¸ ahin, E. Kılı¸c Abstract In this paper, we generalize the geometry of the invariant submanifolds of Riemannian product manifold to the geometry of the invariant submanifolds of Riemannian warped product manifold. We investigate some properties of an invariant submanifolds of a Riemannian warped product manifold. We show that every invariant submanifold of the Riemannian warped product manifold is a Riemannian warped product manifold. Also, we give a theorem on the pseudo-umbilical invariant submanifold. Further, we obtain that integral manifolds on an invariant submanifold are curvature-invariant submanifolds. Finally, we give a necessary condititon on a totally umbilical invariant submanifold to be totally geodesic. Key Words: Riemannian Warped Product Manifold, Vertical and Horizontal Distributions, Pseudo-Umbilical Submanifold, Curvature-Invariant Submanifold. 1. Introduction The geometry of a submanifold (M, g) of a locally product Riemannian manifold (M1 × M2 , g1 ⊗ g2 ) was widely studied by many geometers. In particular, K. Matsumoto has proved that (M , g) is a locally product Riemannian manifold of Riemannian manifolds (M a , ga ) and (M b , gb ), if it is an invariant submanifold of a Riemannian product manifold (M1 × M2 , g1 ⊗ g2 )(see [5]). Later, Xu. Senlin, and Ni. Yilong, ([6]) have updated XMatsumotos and proved that M a ⊂ M1 and M b ⊂ M2 . Moreover, they have proved that (M a , ga ) and (M b , gb ) are pseudo-umbilical submanifolds of (M1 , g1 ) and (M2 , g2 ), respectively, if (M , g) is a pseudo-umbilical submanifold of (M, g) = (M1 × M2 , g1 ⊗ g2 ). They have also demonstrated that M is isometric to the production of its two totally 2000 Mathematics Subject Classification: 53C42, 53C15 407 ˙ KILIC ATC ¸ EKEN, S ¸ AHIN, ¸ geodesic submanifolds (M a , ga ) and (M b , gb ) which are submanifolds of (M1 , g1

• Toán học
• Riemannian warped product manifold
• Vertical and horizontal distributions
• Pseudo umbilical submanifold
• Curvature invariant submanifold
• Totally geodesic
• Semi invariant submanifolds
• Riemannian product
• Warped product
• Locally Riemannian product manifold
• Locally product manifold
• Warped product submanifold
• Skew semi invariant submanifold
• Invariant distribution
• Slant distribution
• Riemannian submersion
• conformal submersion
• Kenmotsu manifold
• Anti invariant
• Semi slant submanifolds
• Semi slant
• Totally umbilical
• Kaehler manifold
• Semiparallel submanifold
• Generalized Sasakian space form
• Laplacian of the second fundamental form
• Totally geodesic submanifold
• Certain conditions
• Anti invariant submersion
• Conformal anti invariant submersion