Abstract
Abstract In this paper we prove two sharp inequalities that relate the normalized scalar curvature with the Casorati curvature for a slant submanifold in a quaternionic space form. Moreover, we show that in both cases, the equality at all points characterizes the invariantly quasi-umbilical submanifolds.
Highlights
In a seminal paper published in the early s, Chen [ ] established a sharp inequality for a submanifold in a real space form using the scalar curvature and the sectional curvature, both being intrinsic invariants, and squared mean curvature, the main extrinsic invariant, initiating the theory of δ-invariants or the so-called Chen invariants; this turned out to be one of the most interesting modern research topic in differential geometry of submanifolds
We note that some interesting inequalities for the length of the second fundamental form of the warped product submanifolds were obtained recently in [ – ]
The Casorati curvature of a submanifold in a Riemannian manifold is an extrinsic invariant defined as the normalized square of the length of the second fundamental form
Summary
In a seminal paper published in the early s, Chen [ ] established a sharp inequality for a submanifold in a real space form using the scalar curvature and the sectional curvature, both being intrinsic invariants, and squared mean curvature, the main extrinsic invariant, initiating the theory of δ-invariants or the so-called Chen invariants; this turned out to be one of the most interesting modern research topic in differential geometry of submanifolds. The Casorati curvature of a submanifold in a Riemannian manifold is an extrinsic invariant defined as the normalized square of the length of the second fundamental form.
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