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A new proof for some optimal inequalities involving generalized normalized δCasorati curvatures
Journal of Inequalities and Applications volume 2015, Article number: 310 (2015)
Abstract
In this paper we give a new proof for two sharp inequalities involving generalized normalized δCasorati curvatures of a slant submanifold in a quaternionic space form. These inequalities were recently obtained in Lee and Vîlcu (Taiwan. J. Math. 19(3):691702, 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.
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 [1]. This theory was initiated in [2]: 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). On the other hand, 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 [3–5]. Some optimal Chenlike inequalities involving Casorati curvatures were proved in [6–10] for several submanifolds in real, complex and quaternionic space forms. Recently, two sharp inequalities involving generalized normalized δCasorati curvatures of slant submanifolds in quaternionic space forms were obtained in [11] as follows.
Theorem 1.1
Let \(M^{n}\) be a θslant proper submanifold of a quaternionic space form \(\overline{M}^{4m}(c)\). Then:

(i)
The generalized normalized δCasorati curvature \(\delta_{C}(r;n1)\) satisfies
$$ \rho\leq\frac{\delta_{C}(r;n1)}{n(n1)}+\frac{c}{4} \biggl(1+ \frac {9}{n1}\cos^{2}\theta \biggr) $$(1)for any real number r such that \(0< r< n(n1)\).

(ii)
The generalized normalized δCasorati curvature \(\widehat{\delta}_{C}(r;n1)\) satisfies
$$ \rho\leq\frac{\widehat{\delta}_{C}(r;n1)}{n(n1)}+\frac{c}{4} \biggl(1+ \frac{9}{n1}\cos^{2}\theta \biggr) $$(2)for any real number \(r>n(n1)\).
Moreover, the equality sign holds in the inequalities (1) and (2) if and only if \(M^{n}\) is an invariantly quasiumbilical submanifold with trivial normal connection in \(\overline{M}^{4m}(c)\), such that with respect to suitable orthonormal tangent frame \(\{\xi _{1},\ldots,\xi_{n}\}\) and normal orthonormal frame \(\{\xi_{n+1},\ldots,\xi _{4m}\}\), the shape operators \(A_{r}\equiv A_{\xi_{r}}\), \(r\in\{n+1,\ldots,4m\}\), take the following forms:
The proof given in [11] for these inequalities is based on an optimization procedure by showing that a quadratic polynomial in the components of the second fundamental form is parabolic. In this work we give an alternative proof using Oprea’s optimization method on submanifolds [12], namely analyzing a suitable constrained extremum problem (see also [13–17]).
Preliminaries
This section gives several basic definitions and notations for our framework based mainly on [18, 19].
Let \(M^{n}\) be an ndimensional Riemannian submanifold of a Riemannian manifold \((\overline{M},\overline{g})\). Then we denote by g the metric tensor induced on M. Let \(K(\pi)\) be the sectional curvature of M associated with a plane section \(\pi\subset T_{p}M\), \(p\in M\). If \(\{e_{1},\ldots,e_{n}\}\) is an orthonormal basis of the tangent space \(T_{p}M\) and \(\{e_{n+1},\ldots,e_{m}\} \) is an orthonormal basis of the normal space \(T_{p}^{\perp}M\), then the scalar curvature τ at p is given by
and the normalized scalar curvature ρ of M is defined as
We denote by H the mean curvature vector, that is,
and we also set
Then it is well known that the squared mean curvature of the submanifold M in M̅ is defined by
and the squared norm of h over dimension n is denoted by \(\mathcal {C}\) and is called the Casorati curvature of the submanifold M. Therefore we have
The submanifold M is called invariantly quasiumbilical if there exist \(mn\) mutually orthogonal unit normal vectors \(\xi _{n+1},\ldots,\xi_{m}\) such that the shape operators with respect to all directions \(\xi_{\alpha}\) have an eigenvalue of multiplicity \(n1\) and that for each \(\xi_{\alpha}\) the distinguished eigendirection is the same.
Suppose now that L is an sdimensional subspace of \(T_{p}M\), \(s\geq 2\), and let \(\{e_{1},\ldots,e_{s}\}\) be an orthonormal basis of L. Then the scalar curvature \(\tau(L)\) of the splane section L is given by
and the Casorati curvature \(\mathcal{C}(L)\) of the subspace L is defined as
The generalized normalized δCasorati curvatures \(\delta _{C}(r;n1)\) and \(\widehat{\delta}_{C}(r;n1)\) of the submanifold \(M^{n}\) are defined for any positive real number \(r\neq n(n1)\) as
if \(0< r< n^{2}n\), and
if \(r>n^{2}n\).
If ∇̅ is the LeviCivita connection on M̅ and ∇ is the covariant differentiation induced on M, then the Gauss and Weingarten formulas are given by
and
where h is the second fundamental form of M, \(\nabla^{\perp}\) is the connection on the normal bundle, and \(A_{N}\) is the shape operator of M with respect to N. If we denote by R̅ and R the curvature tensor fields of ∇̅ and ∇, then we have the Gauss equation:
for all \(X,Y,Z,W\in\Gamma(TM)\).
Assume now that \((\overline{M},\overline{g})\) is a smooth manifold such that there is a rank 3subbundle σ of \(\operatorname{End}(T\overline{M})\) with local basis \(\lbrace{J_{1},J_{2},J_{3}}\rbrace\) satisfying for all \(\alpha\in\{ 1,2,3\}\):
and
where Id denotes the identity tensor field of type \((1, 1)\) on M̅ and the indices are taken from \(\{1,2,3\} \) modulo 3. Then \((\overline{M},\sigma,\overline{g})\) is said to be an almost quaternionic Hermitian manifold. It is easy to see that such a manifold is of dimension 4m, \(m\geq1\). Moreover, if the bundle σ is parallel with respect to the LeviCivita connection ∇̅ of g̅, then \((\overline{M},\sigma,\overline{g})\) is said to be a quaternionic Kähler manifold.
Let \((\overline{M},\sigma,\overline{g})\) be a quaternionic Kähler manifold and let X be a nonnull vector field on M̅. Then the 4plane spanned by \(\{X,J_{1}X,J_{2}X,J_{3}X\}\), denoted by \(Q(X)\), is called a quaternionic 4plane. Any 2plane in \(Q(X)\) is called a quaternionic plane. The sectional curvature of a quaternionic plane is called a quaternionic sectional curvature. A quaternionic Kähler manifold is a quaternionic space form if its quaternionic sectional curvatures are equal to a constant, say c. It is well known that a quaternionic Kähler manifold \((\overline{M},\sigma,\overline{g})\) is a quaternionic space form, denoted \(\overline{M}(c)\), if and only if its curvature tensor is given by
for all vector fields X, Y, Z on M̅ and any local basis \(\lbrace{J_{1},J_{2},J_{3}}\rbrace\) of σ.
A submanifold M of a quaternionic Kähler manifold \((\overline{M},\sigma,\overline{g})\) is said to be a slant submanifold [20] if, for each nonzero vector X tangent to M at p, the angle \(\theta(X)\) between \(J_{\alpha}(X)\) and \(T_{p}M\), \(\alpha\in\{1,2,3\}\) is constant, i.e. it does not depend on the choice of \(p\in M\) and \(X\in T_{p}M\). We can easily see that quaternionic submanifolds are slant submanifolds with \(\theta=0\) and totally real submanifolds are slant submanifolds with \(\theta=\frac{\pi}{2}\). A slant submanifold of a quaternionic Kähler manifold is said to be proper (or θslant proper) if it is neither quaternionic nor totally real. We recall that every proper slant submanifold of a quaternionic Kähler manifold is of even dimension \(n=2s\geq2\) and we can choose a canonical orthonormal local frame, called an adapted slant frame, as follows:
where \(P_{\alpha}e_{2k1}\) denotes the tangential component of \(J_{\alpha}e_{2k1}\), \(k\in\{1,\ldots,s\}\), and α is 1, 2 or 3 (see [21]).
Let \((\overline{M}, \overline{g})\) be a Riemannian manifold, M be a submanifold of M̅, g be the induced metric of g̅ and \(f : M\longrightarrow \mathbb{R}\) be a differentiable function. If we consider the constrained extremum problem
then we have the following result.
Lemma 2.1
[12]
If \(x_{0} \in M\) is the solution of the problem (6), then

(i)
\((\operatorname{grad}(f) )(x_{0}) \in T^{\bot}_{x_{0}}M\);

(ii)
the bilinear form
$$\begin{aligned}& \mathcal{A} : T_{x_{0}}M \times T_{x_{0}}M \longrightarrow \mathbb{R}, \\& \mathcal{A}(X, Y) = \operatorname{Hess}_{f}(X, Y) + \overline{g} \bigl(h(X, Y), \bigl(\operatorname{grad}(f) \bigr) (x_{0})\bigr) \end{aligned}$$
is positive semidefinite, where h is the second fundamental form of M in M̅ and \(\operatorname{grad}(f)\) is the gradient of f.
New proof of Theorem 1.1
Since M is θslant, it is well known from [20] that
where \(P_{\alpha}X\) denotes the tangential component of \(J_{\alpha}X\).
From (7) it follows immediately that
for \(X,Y\in\Gamma(TM)\) and \(\alpha,\beta\in\{1,2,3\}\).
On the other hand, because \(\overline{M}^{4m}(c)\) is a quaternionic space form, from (4) and (5) we derive
Choosing now an adapted slant basis
of \(T_{p}M\), \(p\in M\), where \(2s=n\), from (7) and (8), we derive
and
By using (10) and (11) in (9) we get
We consider now the following quadratic polynomial in the components of the second fundamental form:
where L is a hyperplane of \(T_{p}M\). Without loss of generality we can assume that L is spanned by \(e_{1},\ldots,e_{n1}\). Then we derive
For \(r=n+1, \ldots, 4m\), let us consider the quadratic form \(f_{\alpha} : \mathbb{R}^{n} \longrightarrow\mathbb{R}\) defined by
and the constrained extremum problem
where \(c^{\alpha}\) is a real constant.
The partial derivatives of \(f_{\alpha}\) are
with \(i\in\{1,\ldots,n1\}\), \(i\neq j\), and \(\alpha\in\{n+1,\ldots,4m\}\).
For an optimal solution \(( h_{11}^{\alpha}, \ldots, h_{nn}^{\alpha } )\) of the problem, the vector \(\operatorname{grad}(f_{\alpha})\) is normal at F. That is, it is collinear with the vector \((1, 1, \ldots, 1)\). From (15), it follows that a critical point of the corresponding problem has the form
Using (16) and \(\sum_{i=1}^{n} h_{ii}^{\alpha} = c^{\alpha}\), we derive
We fix an arbitrary point \(x\in F\). The 2form \(\mathcal{A}: T_{x}F \times T_{x}F \longrightarrow\mathbb{R}\) has the form
where h is the second fundamental form of F in \(\mathbb{R}^{n}\) and \(\langle\,,\,\rangle\) is the standard inner product on \(\mathbb{R}^{n}\). Moreover, it is easy to see that the Hessian matrix of \(f_{\alpha}\) has the form
As F is totally geodesic in \(\mathbb{R}^{n}\), considering a vector \(X=(X_{1}, X_{2}, \ldots, X_{n})\) tangent to F at the arbitrary point x on F (that is, verifying the relation \(\sum_{i=1}^{n}X_{i} =0\)), we obtain
Hence the point \(( h_{11}^{\alpha}, \ldots, h_{nn}^{\alpha} )\) from (16) is a global minimum point by Lemma 2.1. Moreover, \(f_{\alpha} ( h_{11}^{\alpha}, \ldots, h_{nn}^{\alpha } ) = 0\). Therefore, we have
and this implies
Therefore we derive
for every tangent hyperplane L of M and both inequalities (1) and (2) obviously follow from (19).
Moreover, we can easily see now that the equality sign holds in the inequalities (1) and (2) if and only if
and
for all \(\alpha\in\{n+1,\ldots,4m\}\).
Finally, from (20) and (21) we deduce that the equality sign holds in (1) and (2) if and only if the submanifold M is invariantly quasiumbilical with trivial normal connection in M̅, such that the shape operators take the forms (3) with respect to suitable tangent and normal orthonormal frames.
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Acknowledgements
The third author was supported by CNCSUEFISCDI, project number PNIIIDPCE201130118.
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Lee, C.W., Lee, J.W. & Vîlcu, G. A new proof for some optimal inequalities involving generalized normalized δCasorati curvatures. J Inequal Appl 2015, 310 (2015). https://doi.org/10.1186/s1366001508310
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MSC
 53A07
 49K35
Keywords
 scalar curvature
 mean curvature
 δCasorati curvature
 shape operator
 quaternionic space form
 slant submanifold
 optimal inequality