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.
In this paper, we obtain a basic Chen's inequality for a Ctotally real submanifold in a generalized (κ, µ)-contact space forms involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvature on the right hand side. Inequalities between the squared mean curvature, k-Ricci curvature and Ricci curvature are also obtained.
In this paper, we introduce a contact pseudo-metric structure on a tangent sphere bundle TεM . we prove that the tangent sphere bundle TεM is (κ, µ)-contact pseudo-metric manifold if and only if the manifold M is of constant sectional curvature. Also, we prove that this structure on the tangent sphere bundle is K-contact iff the base manifold has constant curvature ε.
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