Some Properties on Divine Kaehlerian Manifold

Fibonacci sequence and the divine ratio are intimately inter-connected. In the Fibonacci sequence each number is the sum of previous two consecutive numbers and the ratio of any two consecutive numbers reflects the approximate value of divine ratio. The relationship between divine ratio and Fibonacci series is well express in divergent faunal anatomy and floral as well as their morphology. The present article is intended to study the properties of divine Kaehlerian manifold in terms of Fibonacci sequence, trace & eigen values of divine structure including its almost complex structures. Some properties of induced structures, theorems and propositions related to it have also been studied.

On totally umbilical QR-submanifolds of quaternion Kaehlerian manifolds

We introduce the notion of generalised 3-Sasakian structure on a manifold and show that a totally umbilical, but not totally geodesic, proper QR-submanifold of a quaternion Kaehlerian manifold is an extrinsic sphere and inherits such a structure.

Two theorems on(ϵ)-Sasakian manifolds

In this paper, We prove that every(ϵ)-sasakian manifold is a hypersurface of an indefinite kaehlerian manifold, and give a necessary and sufficient condition for a Riemannian manifold to be an(ϵ)-sasakian manifold.

On the integrability of aK-conformal killing equation in a Kaehlerian manifold

We show that necessary and sufficient condition in order thatK- conformal Killing equation is completely integrable is that the Kaehlerian manifoldK2m(m>2)is of constant holomorphic sectional curvature.

On Totally Real Submanifolds in a 6-Sphere

The 6-dimensional sphere S6 has an almost complex structure induced by properties of Cayley algebra. With respect to this structure S6 is a nearly Kaehlerian manifold. We investigate 2-dimensional totally real submanifolds in S6. We prove that a 2-dimensional totally real submanifold in S6 is flat.

On contact CR-submanifolds of sasakian manifolds

Recently, K.Yano and M.Kon [5] have introduced the notion of a contactCR-submanifold of a Sasakian manifold which is closely similar to the one of aCR-submanifold of a Kaehlerian manifold defined by A. Bejancu [1].In this paper, we shall obtain some fundamental properties of contactCR-submanifolds of a Sasakian manifold. Next, we shall calculate the length of the second fundamental form of a contactCR-product of a Sasakian space form (THEOREM 7.4). At last, we shall prove that a totally umbilical contactCR-submanifold satisfying certain conditions is totally geodesic in the ambient manifold (THEOREM 8.1).