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Remarks on inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms
Journal of Inequalities and Applications volume 2014, Article number: 452 (2014)
Abstract
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.
MSC:53C40, 53D12.
1 Introduction
The Casorati curvature of an n-dimensional submanifold M of a Riemannian manifold, usually denoted by , is an extrinsic invariant defined as the normalized square of the length of the second fundamental form of the submanifold. In [1], Decu et al. introduced the normalized δ-Casorati curvatures and by
and
where , and established some inequalities involving these invariants for submanifolds in real space forms. Later, Slesar et al. proved two inequalities relating the above normalized Casorati curvatures for a slant submanifolds in a quaternionic space form in [2]. However, it was pointed out that the coefficient in (1) is inappropriate and must be replaced by [3, 4]. Following [3, 4], we define the normalized δ-Casorati curvature by
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.
Theorem 1 Let , , be θ-slant proper submanifold of a quaternionic space form . Then the normalized δ-Casorati curvature satisfies
where ρ is the normalized scalar curvature of . Moreover, the equality case holds if and only if is an invariantly quasi-umbilical submanifold with trivial normal connection in , such that with respect to suitable orthonormal tangent frame and normal orthonormal frame , the shape operators , , take the following forms:
2 Preliminaries
Let be an n-dimensional submanifold in an -dimensional Riemannian manifold . The Levi-Civita connections on and will be denoted by and ∇, respectively. For all , , the Gauss and Weingarten formulas can be expressed by
where h is the second fundamental form of M, is the normal connection and the shape operator of M is given by
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].
In we choose a local orthonormal frame , such that, restricting ourselves to , are tangent to . We write . Then the mean curvature vector H is given by
and the squared norm of h over dimension n is denoted by and is called the Casorati curvature of the submanifold M. Therefore we have
Let , , denote the sectional curvature of the plane section spanned by and . Then the scalar curvature of is given by
and the normalized scalar curvature ρ is defined by
Suppose L is an l-dimensional subspace of , , and an orthonormal basis of L. Then the scalar curvature of the l-plane L is given by
and the Casorati curvature of the subspace L is defined as
For more details of slant submanifolds in quaternionic space forms, we refer to [2, 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.
Following [5–7] we considered the constrained extremum problem
then we have the following.
Lemma 1 ([5])
If is the solution of the problem (4), then
-
(i)
;
-
(ii)
the bilinear form
is positive semidefinite, where h is the second fundamental form of in .
In [6], the above lemma was successfully applied to improve an inequality relating obtained in [8]. Later, Chen extended the improved inequality to the general inequalities involving δ-invariants [9]. 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 [16].
4 Proof of Theorem 1
From the Gauss equation we can easily obtain (see (12) in [2])
We define now the following function, denoted by , which is a quadratic polynomial in the components of the second fundamental form:
Without loss of generality, by assuming that L is spanned by , one gets
here we used (5) and (6).
From (7) we have
For , let us consider the quadratic form
and the constrained extremum problem
where is a real constant.
The partial derivatives of the function are
For an optimal solution of the problem in question, the vector is normal at Ϝ, that is, it is collinear with the vector . From (9), (10), (11), and (12), it follows that a critical point of the considered problem has the form
As , by using (13) we have
We fix an arbitrary point . The 2-form has the expression
where is the second fundamental form of Ϝ in and is the standard inner-product on . In the standard frame of , the Hessian of has the matrix
As Ϝ is totally geodesic in , considering a vector X tangent to Ϝ at the arbitrary point x on Ϝ, that is, verifying the relation , we have
Thus the point given by (14) is a global minimum point, here we used Lemma 1. Inserting (14) in (8) we have
From (2), (6), and (15) we can derive inequality (3). The equality case of (3) holds if and only if we have the equality in all the previous inequalities. Thus
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Acknowledgements
We would like to thank to Professor Weidong Song, who has always been generous with his time and advice.
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Zhang, P., Zhang, L. Remarks on inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms. J Inequal Appl 2014, 452 (2014). https://doi.org/10.1186/1029-242X-2014-452
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DOI: https://doi.org/10.1186/1029-242X-2014-452