Abstract. In this paper we study fundamental geometric properties of doubly warped product immersion which is an extension of warped product immersion. Moreover, we study geometric inequality for doubly warped products isometrically immersed in arbitrary Riemannian manifolds. Mathematics Subject Classification (2010). 53C40, 53C42, 53B25.Keywords. doubly warped product, doubly warped immersion, totally umbilical submanifold, shape operator, doubly warped product representaion, geometric inequality, eigenfunction of the Laplacian operator.

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We study Einstein's equation in (m + n)D and (1 + n)D warped spaces (M ,ḡ) and classify all such spaces satisfying Einstein equationsḠ = −Λḡ. We show that the warping function not only can determine the cosmological constantΛ but also it can determine the cosmological constant Λ appearing in the induced Einstein equations G = −Λh on (M 2 , h). Moreover, we discuss on the origin of the 4D cosmological constant as an emergent effect of higher dimensional warped spaces.

Abstract. In this paper we establish a general inequality involving the Laplacian of the warping functions and the squared mean curvature of any doubly warped product isometrically immersed in a Riemannian manifold. Moreover, we obtain some geometric inequalities for C-totally real doubly warped product submanifolds of generalized (κ, µ)-space forms.

The aim is to extend the theory of affine plane curves on R 2 , to the manifolds of 2 and n dimensions. Affine arclength , affine curvatures, and equi-affine vector fields are defined on a manifold of dimension n with equi-affine structure. Classification of equi-affine vector fields on two dimensional equi-affine structure is obtained. Mathematics Subject Classification: 53A15