INEQUALITIES FOR GENERALIZED NORMALIZED $\delta$-CASORATI CURVATURES OF SLANT SUBMANIFOLDS IN QUATERNIONIC SPACE FORMS

Abstract

Abstract. In this paper we prove two sharp inequalities involving the normalized scalar curvature and the generalized normalized δ-Casorati curvatures for slant submanifolds in quaternionic space forms. We also characterize those submanifolds for which the equality cases hold. These results are a generalization of some recent results concerning the Casorati curvature for a slant submanifold in a quaternionic space form obtained by Slesar et al.: J. Inequal. Appl. 2014

Inequalities for casorati curvatures of submanifolds in real space forms

“…The submanifold M n is called to be totally geodesic if h = 0 and minimal if H = 0. Besides, M n is called invariantly quasi-umbilical if there exist p mutually orthogonal unit normal vectors ξ n+1 , · · · , ξ n+p such that the shape operators with respect to all directions ξ r have an eigenvalue of multiplicity n − 1 and that for each ξ r the distinguished eigendirection is the same [15].…”

“…2n [15]. Because the normalized δ−Casorati curvature δ C (n − 1) should be able to be recovered from the generalized normalized δ−Casorati curvature, it would be more appropriate to define δ C (n − 1) using the coefficient n+1 2n .…”

“…Because the normalized δ−Casorati curvature δ C (n − 1) should be able to be recovered from the generalized normalized δ−Casorati curvature, it would be more appropriate to define δ C (n − 1) using the coefficient n+1 2n . More details of generalized normalized δ−Casorati curvature can be found in [15,19]. Following [15], we define the normalized δ−Casorati curvature δ C (n − 1) by…”

“…The proof of Theorem 1.1 is based on an optimization procedure by showing that the quadratic polynomial in the components of the second fundamental form is parabolic. And the above approach was successfully applied to establish inequalities involving Casorati curvatures for different submanifolds in various ambient spaces [15,19,23,24]. In this paper, our main purpose is to present a new approach to establish inequalities for the Casorati curvatures.…”

Optimal Inequalities for the Casorati Curvatures of Submanifolds of Generalized Space Forms Endowed With Semi-Symmetric Metric Connections

“…Therefore, it is of great interest to obtain optimal inequalities for the Casorati curvatures of submanifolds in different ambient spaces. Recently, some optimal inequalities involving Casorati curvatures were proved in [12,13,23,34] for submanifolds in ambient space forms. As a natural prolongation of our research, in this paper we will study these inequalities for submanifolds in generalized space forms, endowed with semi-symmetric metric connections.…”