Remarks on inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms
© Zhang and Zhang; licensee Springer. 2014
Received: 8 August 2014
Accepted: 27 October 2014
Published: 10 November 2014
In this paper, we obtain an inequality for the normalized Casorati curvature of slant submanifolds in quaternionic space forms by using T Oprea’s optimization method.
By using T Oprea’s optimization method on Riemannian submanifolds, we establish the following inequalities in terms of for θ-slant proper submanifolds of a quaternionic space form.
The submanifold M is said to be totally geodesic if . Besides, M is called invariantly quasi-umbilical if there exist p mutually orthogonal unit normal vectors such that the shape operators with respect to all directions have an eigenvalue of multiplicity and that for each the distinguished eigendirection is the same [1–4].
3 Optimization method on Riemannian submanifolds
Let be a Riemannian manifold, be a Riemannian submanifold of it, g be the metric induced on by and be a differentiable function.
then we have the following.
Lemma 1 ()
- (ii)the bilinear form
is positive semidefinite, where h is the second fundamental form of in .
In , the above lemma was successfully applied to improve an inequality relating obtained in . Later, Chen extended the improved inequality to the general inequalities involving δ-invariants . More details of δ-invariants can be found in [10–15]. Besides, the first author gave another proof of the inequalities relating the normalized δ-Casorati curvature for submanifolds in real space forms by using T Oprea’s optimization method .
4 Proof of Theorem 1
here we used (5) and (6).
where is a real constant.
We would like to thank to Professor Weidong Song, who has always been generous with his time and advice.
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