A new proof for some optimal inequalities involving generalized normalized δ-Casorati curvatures
- Chul Woo Lee^{1},
- Jae Won Lee^{2}Email author and
- Gabriel-Eduard Vîlcu^{3, 4}
https://doi.org/10.1186/s13660-015-0831-0
© Lee et al. 2015
Received: 11 May 2015
Accepted: 15 September 2015
Published: 6 October 2015
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.
Keywords
MSC
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 [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 Chen-like 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 δ-Casorati curvatures of slant submanifolds in quaternionic space forms were obtained in [11] as follows.
Theorem 1.1
- (i)The generalized normalized δ-Casorati curvature \(\delta_{C}(r;n-1)\) satisfiesfor any real number r such that \(0< r< n(n-1)\).$$ \rho\leq\frac{\delta_{C}(r;n-1)}{n(n-1)}+\frac{c}{4} \biggl(1+ \frac {9}{n-1}\cos^{2}\theta \biggr) $$(1)
- (ii)The generalized normalized δ-Casorati curvature \(\widehat{\delta}_{C}(r;n-1)\) satisfiesfor any real number \(r>n(n-1)\).$$ \rho\leq\frac{\widehat{\delta}_{C}(r;n-1)}{n(n-1)}+\frac{c}{4} \biggl(1+ \frac{9}{n-1}\cos^{2}\theta \biggr) $$(2)
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]).
2 Preliminaries
This section gives several basic definitions and notations for our framework based mainly on [18, 19].
The submanifold M is called invariantly quasi-umbilical if there exist \(m-n\) 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 \(n-1\) and that for each \(\xi_{\alpha}\) the distinguished eigendirection is the same.
Lemma 2.1
[12]
- (i)
\((\operatorname{grad}(f) )(x_{0}) \in T^{\bot}_{x_{0}}M\);
- (ii)the bilinear form$$\begin{aligned}& \mathcal{A} : T_{x_{0}}M \times T_{x_{0}}M \longrightarrow \mathbb{R}, \\& \mathcal{A}(X, Y) = \operatorname{Hess}_{f}(X, Y) + \overline{g} \bigl(h(X, Y), \bigl(\operatorname{grad}(f) \bigr) (x_{0})\bigr) \end{aligned}$$
3 New proof of Theorem 1.1
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 quasi-umbilical with trivial normal connection in M̅, such that the shape operators take the forms (3) with respect to suitable tangent and normal orthonormal frames.
Declarations
Acknowledgements
The third author was supported by CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0118.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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