Journal of Applied Mathematics and Physics
Vol.06 No.02(2018), Article ID:82641,16 pages
10.4236/jamp.2018.62037
Chen’s Inequalities for Submanifolds in (k, m)-Contact Space Form with a Semi-Symmetric Non-Metric Connection
Asif Ahmad, Faisal Shahzad, Jing Li
Department of Mathematics, Nanjing University of Science and Technology, Nanjing, China
Copyright © 2018 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: January 12, 2018; Accepted: February 24, 2018; Published: February 27, 2018
ABSTRACT
In this paper, we obtain Chen’s inequalities in -contact space form with a semi-symmetric non-metric connection. Also we obtain the inequalites for Ricci and K-Ricci curvatures.
Keywords:
-Contact Space Form, Semi-Symmetric Non-Metric Connection, Chen’s Inequalities, Ricci Curvature
1. Introduction
In 1924, Friedmann and Schouten [1] introduced the idea of a semi-symmetric connection on a differentiable manifold. A linear connection on a differentiable manifold M is said to be semi-symmetric connection if the torsion tensor of the connection satisfies
where is a 1-form.
In 1932, Hayden [2] introduced the notion of a semi-symmetric metric connection on a Riemannian manifold . A semi-symmetric connection is said to be semi-symmetric metric connection if
Yano [3] studied some properties of a Riemannian manifold endowed with a semi-symmetric metric connection. Submanifolds of a Riemannian manifold with a semi-symmetric metric connection were studied by Nakao [4] .
After a long gap, the study of semi-symmetric connection satisfying
(1)
was initiated by Prvanovic [5] with the name Pseudo-metric semi-symmetric connection, and was just followed by Smaranda and Andonie [6] .
A semi-symmetric connection is said to be a semi-symmetric non-metric connection if it satisfies the condition Equation (1).
In 1992, Agashe and Chafle [7] introduced a semi-symmetric non-metric connection on a Riemannian manifold which is given by
where is Riemannian connection on M. They give the relation between the curvature tensor of the manifold with respect to the semi-symmetric non-metric connection and the Riemannian connection. They also proved that the projective curvature tensors of the manifold with respect to these connections are equal to each other.
In 2000, Sengupta, De, and Binh [8] gave another type of semi-symmetric non-metric connection. Özgür [9] studied properties of submanifolds of a Reiemannian manifold with the semi-symmetric non-metric connection.
On the other hand, one of the basic problem in submanifold theory is to find the simple relationship between the intrinsic and extrinsic invariants of a submanifold. Chen [10] [11] [12] , established inequalities in this respect, called Chen inequalities. And many geometers studied similar problems for different submanifolds in various ambient space, see [13] [14] [15] [16] [17] .
Motivated by [7] [21] and [22] , we have studied Chen’s inequalities for submanifolds in -contact space form with a semi-symmetric non-metric connection. The paper is organized as follows. In Section 2, we give a brief introduction about semi-symmetric non-metric connection, -contact space, Chen invarants. In Section 3, for submanifolds in -contact space form with a semi-symmetric non-metric connection we establish the Chen first inequality and Chen Ricci inequalities by using algebraic lemmas.
2. Preliminaries
Let be an -dimensional Riemannian manifold and is a linear connection on . If the torsion tensor
for any vector fields and on satisfies for a 1-form , then the connection is called a semi-symmetric connection.
Let g be a Riemannian metric on . If , then is called a semi-symmetric metric connection on . If , then is called a semi-symmetric non-metric connection on .
Following [7] , a semi-symmetric symmetric non-metric connection on is given by
for any , where denotes the Levi-civita connection with respect to the Riemannian metric g and is a 1-form. Denote by , i.e., the dual vector field U is defined by , for any vector field on .
Let be an n-dimensional submanifold of with the semi-symmetric connection and the Levi-Civita connection . On we consider the induced semi-symmetric connection denoted by and the induced Levi-Civita connection denoted by . The Gauss formula with respect to and can be written as
where is the second fundamental form of and is a -tensor on . According to [18] , we know .
Let and denote the curvature tensor with respect to and respectively. We also denote the curvature tensor and associated with and repectively. From [7] .
(2)
for all , where S is a -tensor field defined by
Denote by the trace of S.
Decomposing the vector field U on M uniquely into its tangent and normal components and , respectively, we have . For any vector field on , the gauss equation with respect to the semi-symmetric non-metric connection is (see [18] )
(3)
In we can choose a local orthonormal frame such that are tangent to . Setting , then the squared lenght of is given by
The mean curvature vector of associated to is . The mean curvature vector of associated to is defined by .
Let be a 2-plane section for any and the sectional curvature of associated to the semi-symmetric non-metric connection . The scalar curvature associated to the semi-symmetric non-metric connection at p is defined by
(4)
Let be a k-plane section of and any orthonormal basis of . The scalar curvature of associated to the semi-symmetric connection is given by
(5)
We denote by . In [12] Chen introduced the first Chen invariant , which is certainly an intrinsic character of .
Suppose L is a k-plane section of and X is a unit vector in L, we choose an orthonormal basis of L, such that . The Ricci curvature of L at X associated to the semi-symmetric metric connection is given by
(6)
where . The is called a K-Ricci curvature. For each integer k, , the Riemannian invariant on is defined by
(7)
where L is a k-plane section in and X is a unit vector in L [19] .
Recently, T. Konfogiorgos intoduced the notion of -contact space form [20] , which contains the well known class of sasakian space forms for . Thus it is worthwhile to study relationships between intrinsic and extrinsic invariants of submanifolds in a -contact space form with a semi-symmetric non-metric connection .
A -dimentional differntiable manifold is called an almost contact metric manifold if there is an almost contact metric structure consisting of a tensor field , a vector field , a 1-form and a compatible Riemannian metric g satisfying
(8)
. An almost contact metric structure becomes a contact metric structure if , where is the fundamental 2-form of .
In a contact metric manifold , the -tensor field h defined by is symmetric and satisfies
The -nullity distribution of a contact metric manifold is a distribution
where k and are constants. If , is called a -contact metric manifold. Since in a -contact metric manifold one has , therefore and if then the structure is Sasakian.
The sectional curvature of a plane section spanned by a unit vector orthogonal to is called a -sectional curvature. If the -contact metric manifold has constant -sectional curvature C, then it is called a -contact space form and it is denoted by . The curvature tensor of is given by [20] .
(9)
, Where if .
For a vector field X on a submanifold M of a -contact form , Let PX be the tangential part of . Thus, P is an endomorphism of the tangent bundle of M and satisfies for . and are the tangential parts of and , respectively. Let be an orthonormal basis of . We set
, . Let be a 2-plane section
spanning by an orthonormal basis . Then given by
is a real number in , which is independent of the choice of orthonormal basis . Put
Then and are also real numbers and do not depend on the choice of orthonormal basis , of course,
3. Chen’s First Inequality
For submanifold of a -contact space form endowed with a semi-symmetric non-matric connection, we establish th following optimal inequality relating the scalar curvature and the squared mean curvature, which will be called Chen first inequality. We recall the following lemma.
Lemma 3.1 ( [22] ) Let for be a function in defined by
If , then we have
with the equality holding if and only if
Theorem 3.1 Let M ba an n-dimensional submanifold of a -dimensional -contact form endowed with a semi-symmetric non-metric connection such that . Then, for each 2-plane section . We have,
(10)
The equality in (10) holds at if and only if there exits an orthonormal basis of and an orthonormal basis of such that (a) and (b) the forms of shape operators
Proof. Let be a 2-plane section. We choose an orthonormal basis for and for such that . Setting , , , . And using (2), (3) and (9) we get
(11)
From (11) we get
(12)
where . On the other hand, using (11) we have
(13)
where is denoted by .
From (12) and (13). It follows that
(14)
Let us consider the following problem:
where is a real constant.
From lemma 3.1, We know
(15)
with the equality holding if and only if
(16)
From (14) and (15), we have
If the equality in (10) holds, then the inequalities given by (14) and (15) become equalities. In this case we have
From [18] we know . So choose a suitable orthonormal basis, the shape operators take the desired forms.
The converse is easy to follow.
For a Sasakian space form , we have and . So using Theorem 3.1, we have the following corollary.
Corollary 3.1 Let M be an n-dimensional submanifold in a sasakian space form endowed with a semi-symmetric non-metric connection such that . Then, for each point and each plane section , we have
(17)
If U is a tangent vector field to M, then the equality in (17) holds at if and only there exists an orthonormal basis of and orthonormal basis of such that
and the forms of shape operators , become
Since in case of non-Sasakian -contact space form, we have , thus and . Putting these values in (17), we can have a direct corollary to Theorem 3.1.
Corollary 3.2 Let Let M be an n-dimensional submanifold in a non-Sasakian -contact space form with a semi-symmetric non-metric connection such that . Then, for each point and each plane section , When , we have
(18)
If U is a tangent vector field to M, then the equality in (18) holds at if and only there exists an orthonormal basis of and orthonormal basis of such that
and the forms of shape operators , become
4. Ricci Curvature and K-Ricci Curvatures
In this section, we establish inequality between Ricci curvature and the squared mean curvature for submanifolds in a -contact space form with a semi-symmetric non-metric connection. This inequality is called Chen-Ricci inequality [19] .
First we give a lemma as following. First we give a lemma as following.
Lemma 4.1 ( [22] ) Let be a function in defined by
If , then we have
with the equality holding if and only if .
Theorem 4.1 Let M be an n-dimensional submanifold of a -dimensional -contact space form endowed with a semi-symmetric non-metric connection such that . Then for each point ,
1) For each unit vector X in , we have
(19)
2) If , a unit tangent vector satisfies the equality case of (19) if and only if .
3) The equality of (19) holds identically for all unit tangent vectors if and only if
either
1) ,
or
2) ,
Proof. (1) Let be an unit vector. We choose an orthonormal basis such that are tangential to M at p with .
Using (11), we have
(20)
Let us consider the function , defined by
We consider the problem
where is a real constant. From lemma 4.1, we have
(21)
With equality holding if and only if
(22)
From (20) and (21) we get
2) For a unit vector , if the equality case of (19) holds, from (20), (21) and (22) we have
Since , we know
So we get
i.e.
The converse is trivial.
3) For all unit vector , the equality case of (19) holds if and only if
Thus we have two cases, namely either or .
In the first case we
In the second case we have
The converse part is straightforward.
Corollary 4.1 Let M be an n-dimensional submanifold in a Sasakian space form endowed with a semi-symmetric non-metric connection such that . Then for each point , For each unit vector X in , for , we have
Corollary 4.2 Let M be an n-dimensional submanifold in a non-Sasakian space form endowed with a semi-symmetric non-metric connection such that . Then for each point , For each unit vector , , we have
Theorem 4.2 Let M be an n-dimensional submanifold in a -dimensional -contact space form endowed with a semi-symmetric non-metric connection such that . Then we have
Proof. Let be an orthonormal basis of . We denote by the k-plane section spanned by . From (5) and (6), it follows that
(23)
and
(24)
Combining (7), (23) and (24), we obtain
(25)
We choose an orthonormal basis of such that is in the direction of the mean curvature vector and diagnolize the shape operator . Then the shape operators take the following forms:
(26)
(27)
From (11), we have
(28)
Using (26) and (28), we obtain
(29)
On the other hand from (26) and (27), we have
(30)
From (29) and (30), it follows that
Using (25), we obtain
Corollary 4.3 Let M be an n-dimensional submanifold in a Sasakian space form endowed with a semi-symmetric non-metric connection such that . Then for each point , For each unit vector , , we have
Corollary 4.4 Let M be an n-dimensional submanifold in a non-Sasakian space form endowed with a semi-symmetric non-metric connection such that . Then for each point , For each unit vector , , we have
Cite this paper
Ahmad, A., Shahzad, F. and Li, J. (2018) Chen’s Inequalities for Submanifolds in (k, m)-Contact Space Form with a Semi-Symmetric Non-Metric Connection. Journal of Applied Mathematics and Physics, 6, 389-404. https://doi.org/10.4236/jamp.2018.62037
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