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An inequality for contact CR-warped product submanifolds of nearly cosymplectic manifolds

Siraj Uddin1* and Khalid Ali Khan2

Author Affiliations

1 Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur, 50603, Malaysia

2 School of Engineering & Logistics, Faculty of Technology, Charles Darwin University, Darwin, NT, 0909, Australia

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Journal of Inequalities and Applications 2012, 2012:304  doi:10.1186/1029-242X-2012-304

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2012/1/304


Received:3 July 2012
Accepted:26 November 2012
Published:18 December 2012

© 2012 Uddin and Khan; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Recently, Chen (Monatshefte Math. 133:177-195, 2001) established general sharp inequalities for CR-warped products in a Kaehler manifold. Afterward, Mihai obtained (Geom. Dedic. 109:165-173, 2004) the same inequalities for contact CR-warped product submanifolds of Sasakian space forms and derived some applications. In this paper, we obtain an inequality for the length of the second fundamental form of the warped product submanifold of a nearly cosymplectic manifold in terms of the warping function. The equality case is also discussed.

MSC: 53C40, 53C42, 53B25.

Keywords:
warped product; contact CR-submanifold; contact CR-warped product; nearly cosymplectic manifold

1 Introduction

An almost contact metric structure <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M1">View MathML</a> satisfying <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M2">View MathML</a> is called a nearly cosymplectic structure. If we consider <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M3">View MathML</a> as a totally geodesic hypersurface of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M4">View MathML</a>, then it is known that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M3">View MathML</a> has a non-cosymplectic nearly cosymplectic structure. Almost contact manifolds with Killing structure tensors were defined in [1] as nearly cosymplectic manifolds, and it was shown that the normal nearly cosymplectic manifolds are cosymplectic (see also [2]). Later on, Blair and Showers [3] studied nearly cosymplectic structure <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M1">View MathML</a> on a manifold <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a> with η closed from the topological viewpoint.

On the other hand, Chen [4] has introduced the notion of CR-warped product submanifolds in a Kaehler manifold. He has established a sharp relationship between the squared norm of the second fundamental form and the warping function. Later on, Mihai [5] studied contact CR-warped products and obtained the same inequality for contact CR-warped product submanifolds isometrically immersed in Sasakian space forms. Motivated by the studies of these authors, many articles dealing with the existence or non-existence of warped products in different settings have appeared; one of them is [3]. In this paper, we obtain an inequality for the length of the second fundamental form in terms of the warping function for contact CR-warped product submanifolds in a more general setting, i.e., nearly cosymplectic manifold.

2 Preliminaries

A <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M8">View MathML</a>-dimensional smooth manifold <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a> is said to have an almost contact structure if on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a> there exist a tensor field ϕ of type <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M11">View MathML</a>, a vector field ξ and a 1-form η satisfying [6]

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M12">View MathML</a>

(2.1)

There always exists a Riemannian metric g on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a> satisfying the following compatibility condition:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M14">View MathML</a>

(2.2)

where X and Y are vector fields on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>[6].

An almost contact structure <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M16">View MathML</a> is said to be normal if the almost complex structure J on the product manifold <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M17">View MathML</a> given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M18">View MathML</a>

where f is a smooth function on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M17">View MathML</a>, has no torsion, i.e., J is integrable, the condition for normality in terms of ϕ, ξ and η is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M20">View MathML</a> on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M22">View MathML</a> is the Nijenhuis tensor of ϕ. Finally, the fundamental 2-form Φ is defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M23">View MathML</a>.

An almost contact metric structure <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M1">View MathML</a> is said to be cosymplectic if it is normal and both Φ and η are closed [6]. The structure is said to be nearly cosymplectic if ϕ is Killing, i.e., if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M25">View MathML</a>

(2.3)

for any X, Y tangent to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M27">View MathML</a> is the Riemannian connection of the metric g on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>. Equation (2.3) is equivalent to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M2">View MathML</a> for each vector field X tangent to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>. The structure is said to be closely cosymplectic if ϕ is Killing and η is closed. It is well known that an almost contact metric manifold is cosymplectic if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M31">View MathML</a> vanishes identically, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M32">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M33">View MathML</a>.

Proposition 2.1[6]

On a nearly cosymplectic manifold, the vector fieldξis Killing.

From the above proposition, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M34">View MathML</a> for any vector field X tangent to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a> is a nearly cosymplectic manifold.

Let M be a submanifold of an almost contact metric manifold <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a> with induced metric g, and let ∇ and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M38">View MathML</a> be the induced connections on the tangent bundle TM and the normal bundle <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M39">View MathML</a> of M, respectively. Denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M40">View MathML</a> the algebra of smooth functions on M and by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M41">View MathML</a> the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M40">View MathML</a>-module of smooth sections of TM over M. Then the Gauss and Weingarten formulas are given by

(2.4)

(2.5)

for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M45">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M46">View MathML</a>, where h and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M47">View MathML</a> are the second fundamental form and the shape operator (corresponding to the normal vector field N), respectively, for the immersion of M into <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>. They are related as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M49">View MathML</a>

(2.6)

where g denotes the Riemannian metric on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a> as well as induced on M.

For any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M51">View MathML</a>, we write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M52">View MathML</a>

(2.7)

where TX is the tangential component and FX is the normal component of ϕX.

A submanifold M tangent to the structure vector field ξ is said to be invariant (resp. anti-invariant) if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M53">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M54">View MathML</a> (resp. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M55">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M54">View MathML</a>).

A submanifold M tangent to the structure vector field ξ of an almost contact metric manifold <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a> is called a contact CR-submanifold (or semi-invariant submanifold) if there exists a pair of orthogonal differentiable distributions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M58">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M59">View MathML</a> on M such that

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M60">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M61">View MathML</a> is the one-dimensional distribution spanned by ξ;

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M58">View MathML</a> is invariant under ϕ, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M63">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M54">View MathML</a>;

(iii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M59">View MathML</a> is anti-invariant under ϕ, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M66">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M54">View MathML</a>.

A contact CR-submanifold is invariant if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M68">View MathML</a> and anti-invariant if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M69">View MathML</a>, respectively. It is called a proper contact CR-submanifold if neither <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M69">View MathML</a> nor <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M68">View MathML</a>. Moreover, if μ is the ϕ-invariant subspace in the normal bundle <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M39">View MathML</a>, then in the case of a contact CR-submanifold, the normal bundle <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M39">View MathML</a> can be decomposed as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M74">View MathML</a>

(2.8)

Bishop and O’Neill [7] introduced the notion of warped product manifolds. They defined these manifolds as follows. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M75">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M76">View MathML</a> be two Riemannian manifolds and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M77">View MathML</a> be a differentiable function on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M78">View MathML</a>. Consider the product manifold <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M79">View MathML</a> with its projections <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M80">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M81">View MathML</a>. Then the warped product of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M78">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M83">View MathML</a> denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M84">View MathML</a> is a Riemannian manifold <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M79">View MathML</a> equipped with the Riemannian structure such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M86">View MathML</a>

for each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M45">View MathML</a> and ⋆ is the symbol for the tangent map. Thus, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M88">View MathML</a>

(2.9)

The function f is called the warping function of the warped product [7]. A warped product <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M89">View MathML</a> is said to be trivial if the warping function f is constant.

We recall the following general result on warped product manifolds for later use.

Lemma 2.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M90">View MathML</a>be a warped product manifold with the warping functionf, then

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M91">View MathML</a>is the lift of<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M92">View MathML</a>on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M78">View MathML</a>,

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M94">View MathML</a>,

(iii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M95">View MathML</a>

for each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M96">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M97">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M98">View MathML</a>is the gradient of the function lnfandand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M99">View MathML</a>denote the Levi-Civita connections onMand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M83">View MathML</a>, respectively.

3 Contact CR-warped product submanifolds

In this section, we consider the warped product submanifolds <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M90">View MathML</a> of a nearly cosymplectic manifold <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M78">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M83">View MathML</a> are Riemannian submanifolds of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>. In the above product, if we assume <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M106">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M107">View MathML</a>, then the warped product of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M78">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M83">View MathML</a> becomes a contact CR-warped product. In this section, we discuss the contact CR-warped products and obtain an inequality for the squared norm of the second fundamental form. For the general case of warped product submanifolds of a nearly cosymplectic manifold, we have the following result.

Theorem 3.1[8]

A warped product submanifold<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M90">View MathML</a>of a nearly cosymplectic manifold<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>is a usual Riemannian product if the structure vector fieldξis tangent to<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M83">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M78">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M83">View MathML</a>are the Riemannian submanifolds of<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>.

If we consider <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M116">View MathML</a>, then for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M117">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M118">View MathML</a>

Taking the inner product with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M117">View MathML</a>, then by Proposition 2.1 and Lemma 2.1(ii), we obtain that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M120">View MathML</a>. This means that either <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M121">View MathML</a>, which is not possible for warped products, or

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M122">View MathML</a>

(3.1)

Now, we consider the warped product contact CR-submanifolds of the types <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M123">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M124">View MathML</a> of a nearly cosymplectic manifold <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>. In [8], the present author has proved that the warped product contact CR-submanifolds of the first type are usual Riemannian products of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127">View MathML</a> are anti-invariant and invariant submanifolds of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>, respectively. In the following, we consider the contact CR-warped product submanifolds <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M124">View MathML</a> and obtain a general inequality. First, we have the following preparatory result for later use.

Lemma 3.1[8]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M132">View MathML</a>be a contact CR-warped product submanifold of a nearly cosymplectic manifold<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M134">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M135">View MathML</a>, then

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M136">View MathML</a>,

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M137">View MathML</a>.

If we replace X by ϕX in (ii) of Lemma 3.1, then we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M138">View MathML</a>

(3.2)

For a Riemannian manifold of dimension m and a smooth function f on M, we recall

(i) ∇f, the gradient of f, is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M139">View MathML</a>

(3.3)

(ii) Δf, the Laplacian of f, is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M140">View MathML</a>

(3.4)

where ∇ is the Levi-Civita connection on M and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M141">View MathML</a> is an orthonormal frame on M.

As a consequence, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M142">View MathML</a>

(3.5)

Now, we prove the main result of this section using the above results.

Theorem 3.2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M132">View MathML</a>be a contact CR-warped product submanifold of a nearly cosymplectic manifold<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>. Then we have

(i) The length of the second fundamental form ofMsatisfies the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M145">View MathML</a>

(3.6)

whereqis the dimension of<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M98">View MathML</a>is the gradient of lnf.

(ii) If the equality sign of (3.6) holds identically, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127">View MathML</a>is a totally geodesic submanifold and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126">View MathML</a>is a totally umbilical submanifold of<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>. Moreover, Mis a minimal submanifold of<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>.

Proof Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a> be a <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M8">View MathML</a>-dimensional nearly cosymplectic manifold and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M132">View MathML</a> be an n-dimensional contact CR-warped product submanifolds of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>. Let us consider the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M156">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M157">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M158">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M159">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M160">View MathML</a> be the local orthonormal frames on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126">View MathML</a>, respectively. Then the orthonormal frames in the normal bundle <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M39">View MathML</a> of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M164">View MathML</a> and μ are <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M165">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M166">View MathML</a>, respectively. Then the length of the second fundamental form h is defined as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M167">View MathML</a>

(3.7)

For the assumed frames, the above equation can be written as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M168">View MathML</a>

(3.8)

The first term on the right-hand side of the above equality is the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M164">View MathML</a>-component and the second term is the μ-component. Here, we equate the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M164">View MathML</a>-component, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M171">View MathML</a>

(3.9)

The above equation can be written for the given frame of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M164">View MathML</a> as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M173">View MathML</a>

Let us decompose the above equation in terms of the components of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M174">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M175">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M176">View MathML</a>, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M177">View MathML</a>

(3.10)

Using Lemma 3.1(i), the first term of (3.10) is identically zero and we shall compute the next term and leave the third term

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M178">View MathML</a>

As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M179">View MathML</a>, then we can write the above equation for one summation, and using (3.2), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M180">View MathML</a>

(3.11)

Using the fact that ξ is tangent to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M182">View MathML</a>, the above equation can be written for the given frame of the distribution <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M58">View MathML</a> as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M184">View MathML</a>

(3.12)

Then from (3.5), the above expression will be

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M185">View MathML</a>

which proves the inequality (3.6). Let us denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M186">View MathML</a>, the second fundamental form of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126">View MathML</a> in M, then by (2.4), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M188">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M189">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M135">View MathML</a>. Thus, on using (3.3), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M191">View MathML</a>

or equivalently,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M192">View MathML</a>

(3.13)

Suppose the equality case holds in (3.6), then from (3.8) and (3.10), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M193">View MathML</a>

(3.14)

As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127">View MathML</a> is a totally geodesic submanifold in M (by Lemma 2.1(i)), using this fact with the first part of (3.14), we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M127">View MathML</a> is totally geodesic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>. On the other hand, the second condition of (3.14) with (3.13) implies that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M126">View MathML</a> is totally umbilical in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>. Moreover, from (3.14), we get M is a minimal submanifold of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/304/mathml/M7">View MathML</a>. This proves the theorem completely. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

SU carried out the whole research and drafted the manuscript. KAK has given the idea of this problem and checked the calculations. All authors read and approved the final manuscript.

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