Abstract
In this paper we give a new proof for two sharp inequalities involving generalized normalized δ-Casorati curvatures of a slant submanifold in a quaternionic space form. These inequalities were recently obtained in Lee and Vîlcu (Taiwan. J. Math. 19(3):691-702, 2015) using an optimization procedure by showing that a quadratic polynomial in the components of the second fundamental form is parabolic. The new proof is obtained analyzing a suitable constrained extremum problem on submanifold.
Highlights
1 Introduction The most powerful tool to find relationships between the main extrinsic invariants and the main intrinsic invariants of a submanifold is provided by Chen’s invariants [ ]. This theory was initiated in [ ]: Chen established a sharp inequality for a submanifold in a real space form using the scalar curvature and the sectional curvature and squared mean curvature
It is well known that 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 and it was preferred by Casorati over the traditional Gauss curvature because corresponds better with the common intuition of curvature [ – ]
It is well known that a quaternionic Kähler manifold (M, σ, g) is a quaternionic space form, denoted M(c), if and only if its curvature tensor is given by c
Summary
The most powerful tool to find relationships between the main extrinsic invariants and the main intrinsic invariants of a submanifold is provided by Chen’s invariants [ ]. Some optimal Chen-like inequalities involving Casorati curvatures were proved in [ – ] for several submanifolds in real, complex and quaternionic space forms. Two sharp inequalities involving generalized normalized δ-Casorati curvatures of slant submanifolds in quaternionic space forms were obtained in [ ] as follows.
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