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The fundamental goal of this study was to achieve the Ricci curvature inequalities for a skew CR-warped product (SCR W-P) submanifold isometrically immersed in a complex space form (CSF) in the expressions of the squared norm of mean curvature vector and warping functions (W-F). The equality cases were likewise examined. In particular, we also derived Ricci curvature inequalities for CR-warped product (CR W-P) submanifolds. To sustain this study, an example of these submanifolds is provided. [ABSTRACT FROM AUTHOR]

In this paper, we show that, given a non-trivial concircular vector field u on a Riemannian manifold (M , g) with potential function f, there exists a unique smooth function ρ on M that connects u to the gradient of potential function ∇ f . We call the connecting function of the concircular vector field u. This connecting function is shown to be a main ingredient in obtaining characterizations of n-sphere S n (c) and the Euclidean space E n . We also show that the connecting function influences on a topology of the Riemannian manifold. [ABSTRACT FROM AUTHOR]