Next Article in Journal
Casimir Physics and Applications
Next Article in Special Issue
On a New type of Tensor on Real Hypersurfaces in Non-Flat Complex Space Forms
Previous Article in Journal
Uncertainty Measurement for a Set-Valued Information System: Gaussian Kernel Method
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications

Department of Mathematics, College of Science, King Khalid University, 9004 Abha, Saudi Arabia
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(2), 200; https://doi.org/10.3390/sym11020200
Submission received: 24 December 2018 / Revised: 23 January 2019 / Accepted: 31 January 2019 / Published: 11 February 2019
(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

Abstract

:
In this paper, by using new-concept pointwise bi-slant immersions, we derive a fundamental inequality theorem for the squared norm of the mean curvature via isometric warped-product pointwise bi-slant immersions into complex space forms, involving the constant holomorphic sectional curvature c, the Laplacian of the well-defined warping function, the squared norm of the warping function, and pointwise slant functions. Some applications are also given.

1. Introduction

In the submanifolds theory, creating a relationship between extrinsic and intrinsic invariants is considered to be one of the most basic problems. Most of these relations play a notable role in submanifolds geometry. The role of immersibility and non-immersibility in studying the submanifolds geometry of a Riemannian manifold was affected by the pioneering work of the Nash embedding theorem [1], where every Riemannian manifold realizes an isometric immersion into a Euclidean space of sufficiently high codimension. This becomes a very useful object for the submanifolds theory, and was taken up by several authors (for instance, see [2,3,4,5,6,7,8,9,10,11,12,13,14,15]). Its main purpose was considered to be how Riemannian manifolds could always be treated as Riemannian submanifolds of Euclidean spaces. Inspired by this fact, Nolker [16] classified the isometric immersions of a warped product decomposition of standard spaces. Motivated by these approaches, Chen started one of his programs of research in order to study the impressibility and non-immersibility of Riemannian warped products into Riemannian manifolds, especially in Riemannian space forms (see [11,17,18,19]). Recently, a lot of solutions have been provided to his problems by many geometers (see [18] and references therein).
The field of study which includes the inequalities for warped products in contact metric manifolds and the Hermitian manifold is gaining importance. In particular, in [17], Chen observed the strong isometrically immersed relationship between the warping function f of a warped product M 1 × f M 2 and the norm of the mean curvature, which isometrically immersed into a real space form.
Theorem 1.
Let M ˜ ( c ) be a m - dimensional real space form and let φ : M = M 1 × f M 2 be an isometric immersion of an n - dimensional warped product into M ˜ ( c ) . Then:
Δ f f n 2 4 n 2 H 2 + n 1 c ,
where n i = d i m M i , i = 1 , 2 , and Δ is the Laplacian operator of M 1 and H is the mean curvature vector of M n . Moreover, the equality holds in (1) if, and only if, φ is mixed and totally geodesic and n 1 H 1 = n 2 H 2 such that H 1 and H 2 are partially mean curvatures of M 1 and M 2 , respectively.
In [2,5,20,21,22,23,24,25,26,27,28,29,30,31], the authors discuss the study of Einstein, contact metrics, and warped product manifolds for the above-mentioned problems. Furthermore, in regard to the collections of such inequalities, we referred to [12] and references therein. The motivation came from the study of Chen and Uddin [32], which proved the non-triviality of warped-product pointwise bi-slant submanifolds of a Kaehler manifold with supporting examples. If the sectional curvature is constant with a Kaehler metric, then it is called complex space forms. In this paper, we consider the warped-product pointwise bi-slant submanifolds which isometrically immerse into a complex space form, where we then obtain a relationship between the squared norm of the mean curvature, constant sectional curvature, the warping function, and pointwise bi-slant functions. We will announce the main result of this paper in the following.
Theorem 2.
Let M ˜ 2 m ( c ) be the complex space form and let φ : M n = M 1 n 1 × f M 2 n 2 M ˜ 2 m ( c ) be an isometric immersion from warped product pointwise bi-slant submanifolds into M ˜ 2 m ( c ) . Then, the following inequality is satisfied:
Δ ( l n f ) ln f 2 + n 2 4 n 2 H 2 + n 1 c 4 3 c 4 n 2 n 1 cos 2 θ 1 + n 2 cos 2 θ 2 ,
where θ 1 and θ 2 are pointwise slant functions along M 1 and M 2 , respectively. Furthermore, and Δ are the gradient and the Laplacian operator on M 1 n 1 , respectively, and H is the mean curvature vector of M n . The equality case holds in (2) if and only if φ is a mixed totally geodesic isometric immersion and the following satisfies
H 1 H 2 = n 2 n 1
where H 1 and H 2 are the mean curvature vectors along M 1 n 1 and M 2 n 2 , respectively.
As an application of Theorem 2 in a compact orientated Riemannian manifold with a free boundary condition, we prove that:
Theorem 3.
Let M n = M 1 n 1 × f M 2 n 2 be a compact, orientate warped product pointwise bi-slant submanifold in a complex space form M ˜ 2 m ( c ) such that M 1 n 1 is a n 1 -dimensional and M 2 n 2 is a n 2 - dimensional pointwise slant submanifold M ˜ 2 m ( c ) . Then, M n is simply a Riemannian product if, and only if:
H 2 c n 2 3 n 1 cos 2 θ 1 + 3 n 2 cos 2 θ 2 n 1 n 2 ,
where H is the mean curvature vector of M n . Moreover, θ 1 and θ 2 are pointwise slant functions.
By using classifications of pointwise bi-slant submanifolds which were defined in [32], we derived similar inequalities for warped product pointwise pseudo-slant submanifolds [33], warped product pointwise semi-slant submanifolds [34], and CR-warped product submanifolds [17] in a complex space form as well.

2. Preliminaries and Notations

An almost complex structure J and a Riemannian metric g, such that J 2 = I and g ( J X , J Y ) = g ( X , Y ) , for X , Y X ( M ˜ ) , where I denotes the identity map and X ( M ˜ ) is the space containing vector fields tangent to M ˜ , then ( M , J , g ) is an almost Hermitian manifold. If the almost complex structure satisfied ( ˜ U J ) V = 0 , for any U , V X ( M ˜ ) and ˜ is a Levi-Cevita connection M ˜ . In this case, M ˜ is called the Kaehler manifold. A complex space form of constant holomorphic sectional curvature c is denoted by M ˜ 2 m ( c ) , and its curvature tensor R ˜ can be expressed as:
R ˜ U , V , Z , W = c 4 ( g ( U , Z ) g ( V , W ) g ( V , Z ) g ( U , W ) + g ( U , J Z ) g ( J V , W ) g ( V , J Z ) g ( U , J W ) + 2 g ( U , J V ) g ( J Z , W ) ) ,
for every U , V , Z , W X ( M ˜ 2 m ( c ) ) . A Riemannian manifold M ˜ m and its submanifold M, the Gauss and Weingarten formulas are defined by ˜ U V = U V + h ( U , V ) , a n d ˜ U ξ = A ξ U + U ξ , respectively for each U , V X ( M ) and for the normal vector field ξ of M, where h and A ξ are denoted as the second fundamental form and shape operator. They are related as g ( h ( U , V ) , N ) = g ( A N U , V ) . Now, for any U X ( M ) and for the normal vector field ξ of M, we have:
( i ) J U = P U + F U , ( i i ) J ξ = t ξ + f ξ ,
where P U ( t ξ ) and F U ( f ξ ) are tangential to M and normal to M, respectively. Similarly, the equations of Gauss are given by:
R ( U , V , Z , W ) = R ˜ ( U , V , Z , W ) + g h ( U , W ) , h ( V , Z ) g h ( U , Z ) , h ( V , W ) .
for all U , V , Z , W are tangent M, where R and R ˜ are defined as the curvature tensor of M ˜ m and M n , respectively.
The mean curvature H of Riemannian submanifold M n is given by
H = 1 n t r a c e ( h ) .
A submanifold M n of Riemannian manifold M ˜ m is said to be totally umbilical and totally geodesic if h ( U , V ) = g ( U , V ) H and h ( U , V ) = 0 , for any U , V X ( M ) , respectively, where H is the mean curvature vector of M n . Furthermore, if H = 0 , them M n is minimal in M ˜ m .
A new class called a “pointwise slant submanifold” has been studied in almost Hermitian manifolds by Chen-Gray [35]. They provided the following definitions of these submanifolds:
Definition 1.
[35] A submanifold M n of an almost Hermitian manifold M ˜ 2 m is a pointwise slant if, for any non-zero vector X X ( T x M ) and each given point x M n , the angle θ ( X ) between J X and tangent space T x M is free from the choice of the nonzero vector X. In this case, the Wirtinger angle become a real-valued function and it is non-constant along M n , which is defined on T * M such that θ : T * M R .
Chen-Gray in [35] derived a characterization for the pointwise slant submanifold, where M n is a pointwise slant submanifold if, and only if, there exists a constant λ [ 0 , 1 ] such that P 2 = cos 2 θ I , where P is a (1,1) tensor field and I is an identity map. For more classifications, we referred to [35].
Following the above concept, a pointwise bi-slant immersion was defined by Chen-Uddin in [18], where they defined it as follows:
Definition 2.
A submanifold M n of an almost Hermitian manifold M ˜ 2 m is said to be a pointwise bi-slant submanifold if there exists a pair of orthogonal distributions D θ 1 and D θ 2 , such that:
(i) 
T M n = D θ 1 D θ 2 ;
(ii) 
J D θ 1 D θ 2 a n d J D θ 2 D θ 1 ;
(iii) 
Each distribution D θ i is a pointwise slant with a slant function θ i : T * M R f o r i = 1 , 2 .
Remark 1.
A pointwise bi-slant submanifold is a bi-slant submanifold if each slant functions θ i : T * M R f o r i = 1 , 2 . are constant along M n (see [13]).
Remark 2.
If θ 1 = π 2 o r θ 2 = π 2 , then M n is called a pointwise pseudo-slant submanifold (see [33]).
Remark 3.
If θ 1 = 0 o r θ 2 = 0 , in this case, M n is a coinciding pointwise semi-slant submanifold (see [14,34]).
Remark 4.
If θ 2 = π 2 and θ 1 = 0 , then M n is CR-submanifold of the almost Hermitian manifold.
In this context, we shall define another important Riemannian intrinsic invariant called the scalar curvature of M ˜ m , and denoted at τ ˜ ( T x M ˜ m ) , which, at some x in M ˜ m , is given:
τ ˜ ( T x M ˜ m ) = 1 α < β m K ˜ α β ,
where K ˜ α β = K ˜ e α e β . It is clear that the first equality (7) is congruent to the following equation, which will be frequently used in subsequent proof:
2 τ ˜ ( T x M ˜ m ) = 1 α < β m K ˜ α β , 1 α , β n .
Similarly, scalar curvature τ ˜ ( L x ) of L - plan is given by:
τ ˜ ( L x ) = 1 α < β m K ˜ α β ,
An orthonormal basis of the tangent space T x M is { e 1 , e n } such that e r = ( e n + 1 , e m ) belong to the normal space T M . Then, we have:
h α β r = g ( h ( e α , e β ) , e r ) , h 2 = α , β = 1 n g h ( e α , e β ) , h ( e α , e β .
Let K α β and K ˜ α β be the sectional curvatures of the plane section spanned by e α and e β at x in a submanifold M n and a Riemannian manifold M ˜ m , respectively. Thus, K α β and K ˜ α β are the intrinsic and extrinsic sectional curvatures of the span { e α , e β } at x. Thus, from the Gauss Equation (6)(i), we have:
2 τ ( T x M n ) = K α β = 2 τ ˜ ( T x M n ) + r = n + 1 m h α α r h β β r ( h α β r ) 2 = K ˜ α β + r = n + 1 m h α α r h β β r ( h α β r ) 2 .
The following consequences come from (6) and (11), as:
τ ( T x M 1 n 1 ) = r = n + 1 m 1 i < j n 1 h i i r h j j r ( h i j r ) 2 + τ ˜ ( T x M 1 n 1 ) .
Similarly, we have:
τ ( T x M 2 n 2 ) = r = n + 1 m n 1 + 1 a < b n h a a r h b b r ( h a b r ) 2 + τ ˜ ( T x M 2 n 2 ) .
Assume that M 1 n 1 and M 2 n 2 are two Riemannian manifolds with their Riemannian metrics g 1 and g 2 , respectively. Let f be a smooth function defined on M 1 n 1 . Then, the warped product manifold M n = M 1 n 1 × f M 2 n 2 is the manifold M 1 n 1 × M 2 n 2 furnished by the Riemannian metric g = g 1 + f 2 g 2 , which defined in [36]. When considering that the M n = M 1 n 1 × f M 2 n 2 is the warped product manifold, then for any X X ( M 1 ) and Z X ( M 2 ) , we find that:
Z X = X Z = ( X ln f ) Z .
Let { e 1 , e n } be an orthonormal frame for M n ; then, summing up the vector fields such that:
i = 1 n 1 j = 1 n 2 K ( e α e β ) = α = 1 n 1 β = 1 n 2 e α e α ln f e α e β ln f e α ln f 2 .
From (Equation (3.3) in [11]), the above equation implies that:
α = 1 n 1 β = 1 n 2 K ( e α e β ) = n 2 Δ ( ln f ) ( ln f ) 2 = n 2 Δ f f .
Remark 5.
A warped product manifold M n = M 1 n 1 × f M 2 n 2 is said to be trivial or a simple Riemannian product manifold if the warping function f is constant.

3. Main Inequality for Warped Product Pointwise Bi-Slant Submanifolds

To obtain similar inequalities like Theorem 1, for warped product pointwise bi-slant submanifolds of complex space forms, we need to recall the following lemma.
Lemma 1.
[10] Let a 1 , a 2 , a n , a n + 1 be n + 1 be real numbers with
( i = 1 n a i ) 2 = ( n 1 ) ( i = 1 n a i 2 + a n + 1 ) , n 2 .
Then 2 a 1 . a 2 a 3 holds if and only if a 1 + a 2 = a 3 = = a k .
Proof of Theorem 2.
If substitute X = Z = e α and Y = W = e β for 1 α , β n in (4), and (6), taking summing up then
α , β = 1 n R ˜ ( e α , e β , e α , e β ) = c 4 n ( n 1 ) + 3 α , β = 1 n g 2 ( J e α , e β ) .
As M n is a pointwise bi-slant submanifold, we defined an adapted orthonormal frame as n = 2 d 1 + 2 d 2 follows e 1 , e 2 = sec θ 1 P e 1 , , e 2 d 1 1 , e 2 d 1 = sec θ 1 P e 2 d 1 1 , , e 2 d 1 + 1 , e 2 d 1 + 2 = sec θ 2 P e 2 d 1 + 1 , , e 2 d 1 + 2 d 2 1 , e 2 d 1 + 2 d 2 = sec θ 2 P e 2 d 1 + 2 d 2 1 . Thus, we defined it such that g ( e 1 , J e 2 ) = g ( J e 1 , e 2 ) = g ( J e 1 , sec θ 1 P e 1 ) , which implies that g ( e 1 , J e 2 ) = sec θ 1 g ( P e 1 , P e 1 ) . □
Following ((2.8) in [32]), we get g ( e 1 , J e 2 ) = cos θ 1 g ( e 1 , e 2 ) . Therefore, we easily obtained the following relation:
g 2 ( e α , J e β ) = cos 2 θ 1 , f o r e a c h α = 1 , , 2 d 1 1 , cos 2 θ 2 , f o r e a c h β = 2 d 1 + 1 , , 2 d 1 + 2 d 1 1 .
Hence, we have:
α , β = 1 n g 2 ( J e α , e β ) = n 1 cos 2 θ + n 2 cos 2 θ .
Following from (17), (16), and (6), we find that:
2 τ = c 4 n ( n 1 ) + c 4 3 n 1 cos 2 θ 1 + 3 n 2 cos 2 θ 2 + n 2 H 2 h 2 .
Let us assume that:
δ = 2 τ c 4 n ( n 1 ) c 4 3 n 1 cos 2 θ 1 + 3 n 2 cos 2 θ 2 n 2 2 H 2 .
Then, from (19), and (18), we get:
n 2 H 2 = 2 δ + h 2 .
Thus, from an orthogonal frame { e 1 , e 2 , e n } , the proceeding equation takes the new form:
r = n + 1 2 m i = 1 n h A A r 2 = 2 ( δ + r = n + 1 2 m i = 1 n ( h A A r ) 2 + r = n + 1 2 m + 1 i < j = 1 n ( h A B r ) 2 + r = n + 1 2 m A , B = 1 n ( h A B r ) 2 ) .
This can be expressed in more detail, such as:
1 2 h 11 n + 1 + A = 2 n 1 h A A n + 1 + l = n 1 + 1 n h l l n + 1 2 = δ + ( h 11 n + 1 ) 2 + A = 2 n 1 ( h A A n + 1 ) 2 + l = n 1 + 1 n ( h l l n + 1 ) 2 2 B q n 1 h B B n + 1 h q q n + 1 n 1 + 1 l s n h l l n + 1 h s s n + 1 + A < B = 1 n ( h A B n + 1 ) 2 + r = n + 1 2 m A , B = 1 n ( h A B r ) 2 .
Assume that a 1 = h 11 n + 1 , a 2 = A = 2 n 1 h A A n + 1 , and a 3 = l = n 1 + 1 n h l l n + 1 . Then, applying Lemma 1 in (22), we derive:
δ 2 + A < B = 1 n ( h A B n + 1 ) 2 + 1 2 r = n + 1 2 m A , B = 1 n ( h A B r ) 2 2 B q n 1 h B B n + 1 h q q n + 1 + n 1 + 1 l s n h l l n + 1 h s s n + 1 .
with equality holds in (23) if and only if
A = 2 n 1 h A A n + 1 = B = n 1 + 1 n h B B n + 1 .
On the other hand, from (15), we have:
n 2 Δ f f = τ 1 A < B n 1 K ( e A e B ) n 1 + 1 l < q n K ( e l e q ) .
Then from (6) and the scalar curvature for the complex space form (11), we get:
n 2 Δ f f = τ n 1 ( n 1 1 ) c 8 3 n 1 c 4 cos 2 θ 1 r = n + 1 2 m 1 A B n 1 ( h A A r h B B r ( h A B r ) 2 ) n 2 ( n 2 1 ) c 8 3 n 2 c 4 cos 2 θ 2 r = n + 1 2 m n 1 + 1 l q n ( h l l r h q q r ( h l q r ) 2 ) .
Now from (23) and (26), we have:
n 2 Δ f f ρ n ( n 1 ) c 8 + n 1 n 2 c 4 3 n 1 c 4 cos 2 θ 1 δ 2 3 n 2 c 4 cos 2 θ 2 .
Using (19) in the above equation and relation Δ f f = Δ ( ln f ) ln f 2 , we derive:
n 2 Δ ( ln f ) ln f 2 n 2 4 H 2 + c 4 n 1 n 2 + 3 n 1 cos 2 θ 1 + 3 n 2 cos 2 θ 2 .
which implies inequality. The equality sign holds in (2) if, and only if, the leaving terms in (23) and (24) imply that:
r = n + 2 2 m B = 1 n 1 h B B r = r = n + 2 2 m A = n 1 + 1 n 1 h A A r = 0 ,
and n 1 H 1 = n 2 H 2 , where H 1 and H 2 are partially mean curvature vectors on M 1 n 1 and M 2 n 2 , respectively. Moreover, also from (23), we find that
h A B r = 0 , f o r e a c h 1 A n 1 n 1 + 1 B n n + 1 r 2 m .
This shows that φ is a mixed, totally geodesic immersion. The converse part of (30) is true in a warped product pointwise bi-slant into the complex space form. Thus, we reached our promised result.

Consequences of Theorem 2

Inspired by the research in [6,34] and using the Remark 3 in Theorem 2 for pointwise semi-slant warped product submanifolds, we obtained:
Corollary 1.
Let φ : M n = M 1 n 1 × f M 2 n 2 M ˜ 2 m ( c ) be an isometric immersion from the warped product pointwise semi-slant submanifold into a complex space form M ˜ 2 m ( c ) , where M 1 n 1 is the holomorphic and M 2 n 2 is the pointwise slant submanifolds of M ˜ 2 m ( c ) . Then, we have the following inequality:
Δ ( l n f ) ln f 2 + n 2 4 n 2 H 2 + n 1 c 4 3 c 4 n 2 n 1 + n 2 cos 2 θ ,
where n i = d i m M i , i = 1 , 2 . Furthermore, and Δ are the gradient and the Laplacian operator on M 1 n 1 , respectively, and H is the mean curvature vector of M n . The equality sign holds in (31) if, and only if, n 1 H 1 = n 2 H 2 , where H 1 and H 2 are the mean curvature vectors along M 1 n 1 and M 2 n 2 , respectively, and φ is a mixed, totally geodesic immersion.
From the motivation studied in [14,34], we present the following consequence of Theorem 2 by using the Remark 2 for a nontrivial warped product pointwise pseudo-slant submanifold of a complex space, such that:
Corollary 2.
Let φ : M n = M 1 n 1 × f M 2 n 2 M ˜ 2 m ( c ) be an isometric immersion from a warped product pointwise pseudo-slant submanifold into a complex space form M ˜ 2 m ( c ) , such that M 1 n 1 is a totally real and M 2 n 2 is a pointwise slant submanifold of M ˜ 2 m ( c ) . Then, we have the following inequality:
Δ ( l n f ) ln f 2 + n 2 4 n 2 H 2 + n 1 c 4 3 c 4 cos 2 θ ,
where n i = d i m M i , i = 1 , 2 . Furthermore, and Δ are the gradient and the Laplacian operator on M 1 n 1 , respectively, and H is the mean curvature vector of M n . The equality condition holds in (32) if, and only if, the following satisfies
H 1 H 2 = n 2 n 1
: where H 1 and H 2 are the mean curvature vectors along M 1 n 1 and M 2 n 2 , respectively, and φ is a mixed, totally geodesic isometric immersion.
Corollary 3.
Let φ : M n = M 1 n 1 × f M 2 n 2 M ˜ 2 m ( c ) be an isometric immersion from a warped product pointwise pseudo-slant submanifold into a complex space form M ˜ 2 m ( c ) , such that M 1 n 1 is a pointwise slant and M 2 n 2 is a totally real submanifold of M ˜ 2 m ( c ) . Then, we have the following:
Δ ( l n f ) ln f 2 + n 2 4 n 2 H 2 + n 1 c 4 3 n 1 c 4 n 2 cos 2 θ ,
where n i = d i m M i , i = 1 , 2 . Furthermore, and Δ are the gradient and the Laplacian operator on M 1 n 1 , respectively, and H is the mean curvature vector of M n . This equally holds in (33) if, and only if, φ is a mixed, totally geodesic isometric immersion and the following satisfies
H 1 H 2 = n 2 n 1 ,
, where H 1 and H 2 are the mean curvature vectors along M 1 n 1 and M 2 n 2 , respectively.
Similarly, using Remark 4 and from [17], we got the following result from Theorem 2:
Corollary 4.
Let φ : M n = M 1 n 1 × f M 2 n 2 M ˜ 2 m ( c ) be an isometric immersion from a CR-warped product into a complex space form M ˜ 2 m ( c ) , such that M 1 n 1 is a holomorphic submanifold and M 2 n 2 is a totally real submanifold of M ˜ 2 m ( c ) . Then, we get the following:
Δ ( l n f ) ln f 2 + n 2 4 n 2 H 2 + n 1 c 4 3 n 1 c 4 n 2 ,
where n i = d i m M i , i = 1 , 2 . Furthermore, and Δ are the gradient and the Laplacian operator on M 1 n 1 , respectively, and H is the mean curvature vector of M n . The same holds in (34) if, and only if, φ is mixed and totally geodesic, and n 1 H 1 = n 2 H 2 , where H 1 and H 2 are the mean curvature vectors on M 1 n 1 and M 2 n 2 , respectively.
In particular, if both pointwise slant functions θ 1 = θ 2 = π 2 , then M n is becomes a totally real warped product submanifold—thus, we obtain:
Corollary 5.
Let φ : M n = M 1 n 1 × f M 2 n 2 M ˜ 2 m ( c ) be an isometric immersion from an n-dimensional, totally real warped product submanifold into a 2 m -dimensional complex space form M ˜ 2 m ( c ) , where M 1 n 1 and M 2 n 2 are totally real submanifolds of M ˜ 2 m ( c ) . Then, we have the following:
Δ ( l n f ) ln f 2 + n 2 4 n 2 H 2 + n 1 c 4 ,
where n i = d i m M i , i = 1 , 2 and Δ is the Laplacian operator on M 1 n 1 . The same holds in (35) if, and only if, φ is mixed and totally geodesic, and the following satisfies
H 1 H 2 = n 2 n 1 ,
where H 1 and H 2 are the mean curvature vectors on M 1 n 1 and M 2 n 2 , respectively.
Proof of Theorem 3.
In this direction, we consider the warped product pointwise bi-slant submanifolds as a compact oriented Riemannian manifold without boundary. If the inequality (2) holds:
Δ ( l n f ) ln f 2 n 2 4 n 2 H 2 + c 4 n 2 n 1 n 2 3 n 1 cos 2 θ 1 3 n 2 cos 2 θ 2 .
Since M n is a compact oriented Riemannian submanifold without boundary, then we have following formula with respect to the volume element:
M n Δ f d V = 0 .
From the hypothesis of the theorem, M n is a compact warped product submanifold; then from (37), we derive:
M c 4 n 2 3 n 1 cos 2 θ 1 + 3 n 2 cos 2 θ 2 n 1 n 2 1 4 n 2 i = 1 n ( h i i n + 1 ) 2 d V M ( ln f 2 ) d V .
Now, we assume that M n is a Riemannian product, and the warping function f must be constant on M n . Then, from (38), we get the inequality (3). □
Conversely, let the inequality (3) hold; then from (38), we derive:
0 M n ( ln f 2 ) 0 .
The above condition implies that ln f 2 = 0 , where this means that f is a constant function on M n . Hence, M n is simply a Riemannian product of M 1 n 1 and M 2 n 2 , respectively. Thus, the theorem is proved. We give some other important corollaries as consequences of Theorem 2, as follows:
Corollary 6.
Let M n = M 1 n 1 × f M 2 n 2 be a warped product pointwise bi-slant submanifold of a complex space form M ˜ 2 m ( c ) with warping function f, such that n 1 = d i m M 1 and n 2 = d i m M 2 . If φ is an isometrically minimal immersion from warped product M n into M ˜ 2 m ( c ) , then we obtain:
Δ ( l n f ) ln f 2 + c 4 n 2 n 1 n 2 3 n 1 cos 2 θ 1 3 n 2 cos 2 θ 2 .
Corollary 7.
Let M n = M 1 n 1 × f M 2 n 2 be a warped product pointwise bi-slant submanifold of a complex space form M ˜ 2 m ( c ) with warping function f, such that n 1 = d i m M 1 and n 2 = d i m M θ . Then, there is no existing minimal isometric immersion φ from warped product M n into M ˜ 2 m ( c ) with:
Δ ( l n f ) > ln f 2 + c 4 n 2 n 1 n 2 3 n 1 cos 2 θ 1 3 n 2 cos 2 θ 2 .

Author Contributions

All authors made an equal contribution to draft the manuscript.

Funding

The authors extended their appreciation to the Deanship of Scientific Research at King Khalid University, for funding this work through research groups program under grant number R.G.P.1/79/40.

Acknowledgments

The authors would like to thank the referees for their stick criticism and suggestions on this paper to improve the quality. The authors are grateful to Siraj Uddin for his useful comments, discussions and constant encouragement which improved the paper.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Nash, J. The imbedding problem for Riemannian manifolds. Ann. Math. 1956, 63, 20–63. [Google Scholar] [CrossRef]
  2. Ali, A.; Lee, J.W.; Alkhaldi, A.H. Geometric classification of warped product submanifolds of nearly Kaehler manifolds with a slant fiber. Int. J. Geom. Methods. Mod. Phys. 2018. [Google Scholar] [CrossRef]
  3. Ali, A.; Laurian-Ioan, P. Geometric classification of warped products isometrically immersed in Sasakian space forms. Math. Nachr. 2018, 292, 234–251. [Google Scholar]
  4. Ali, A.; Laurian-Ioan, P. Geometry of warped product immersions of Kenmotsu space forms and its applications to slant immersions. J. Geom. Phys. 2017, 114, 276–290. [Google Scholar] [CrossRef]
  5. Ali, A.; Ozel, C. Geometry of warped product pointwise semi-slant submanifolds of cosymplectic manifolds and its applications. Int. J. Geom. Methods Mod. Phys. 2017, 14, 175002. [Google Scholar] [CrossRef]
  6. Ali, A.; Uddin, S.; Othmam, W.A.M. Geometry of warped product pointwise semi-slant submanifold in Kaehler manifolds. Filomat 2017, 31, 3771–3788. [Google Scholar] [CrossRef]
  7. Chen, B.-Y. A general inequality for submanifolds in complex-space-forms and its applications. Arch. Math. 1996, 67, 519–528. [Google Scholar] [CrossRef]
  8. Chen, B.-Y. Mean curvature and shape operator of isometric im-mersions in real-space-forms. Glasgow Math. J. 1996, 38, 87–97. [Google Scholar] [CrossRef]
  9. Chen, B.-Y. Relations between Ricci curvature and shape operatorfor submanifolds with arbitrary codimension. Glasgow Math. J. 1999, 41, 33–41. [Google Scholar] [CrossRef]
  10. Chen, B.-Y.; Dillen, F.; Verstraelen, L.; Vrancken, L. Characterization of Riemannian space forms, Einstein spaces and conformally flate spaces. Proc. Am. Math. Soc. 1999, 128, 589–598. [Google Scholar] [CrossRef]
  11. Chen, B.-Y. On isometric minimal immersions from warped products into real space forms. Proc. Edinb. Math. Soc. 2002, 45, 579–587. [Google Scholar] [CrossRef]
  12. Chen, B.-Y. Pseudo-Riemannian Geometry, δ-Invariants and Applications; World Scientific: Hackensack, NJ, USA, 2011. [Google Scholar]
  13. Uddin, S.; Chen, B.-Y.; Al-Solamy, F.R. Warped product bi-slant immersions in Kaehler manifolds. Mediterr. J. Math. 2017, 14, 95. [Google Scholar] [CrossRef]
  14. Uddin, S.; Stankovic, M.S. Warped product submanifolds of Kaehler manifolds with pointwise slant fiber. Filomat 2018, 32, 35–44. [Google Scholar] [CrossRef]
  15. Uddin, S.; Al-Solamy, F.R.; Shahid, M.H.; Saloom, A.B.-Y. Chen’s inequality for bi-warped products and its applications in Kenmotsu manifolds. Mediterr. J. Math. 2018, 15, 193. [Google Scholar] [CrossRef]
  16. Nolker, S. Isometric immersions of warped products. Differ. Geom. Appl. 1996, 6, 1–30. [Google Scholar] [CrossRef]
  17. Chen, B.-Y. Geometry of warped product CR-submanifolds in Kaehler manifolds. Monatsh. Math. 2001, 133, 177–195. [Google Scholar] [CrossRef]
  18. Chen, B.-Y. Differential Geometry of Warped Product Manifolds and Submanifolds; World Scientific: Hackensack, NJ, USA, 2017. [Google Scholar]
  19. Uddin, S.; Al-Solamy, F.R. Warped product pseudo-slant immersions in Sasakian manifolds. Publ. Math. Debrecen 2017, 91, 331–348. [Google Scholar] [CrossRef]
  20. Al-Solamy, F.R.; Khan, V.A.; Uddin, S. Geometry of warped product semi-slant submanifolds of nearly Kaehler manifolds. Results Math. 2017, 71, 783–799. [Google Scholar] [CrossRef]
  21. Alqahtani, L.S.; Uddin, S. Warped product pointwise pseudo-plant submanifolds of locally product Riemannian manifolds. Filomat 2018, 32, 423–438. [Google Scholar] [CrossRef]
  22. Chen, B.-Y. A general optimal inequality for warped products in complex projective spaces and its applications. Proc. Jpn. Acad. Ser. A 2003, 79, 89–94. [Google Scholar] [CrossRef]
  23. Defever, F.; Mihai, I.; Verstraelen, L. B. Y. Chen’s inequality for C-totally real submanifolds in Sasakian space forms. Boll. Unione Matematica Ital. B 1997, 11, 365–374. [Google Scholar]
  24. Decu, S.; Haesen, S.; Verstraelen, L.; Vîlcu, G.E. Curvature invariants of Statistical Submanifolds in Kenmotsu Statistical manifolds of constant φ-sectional curvature. Entropy 2018, 20, 529. [Google Scholar] [CrossRef]
  25. He, G.; Liu, H.; Zhang, L. Optimal inequalities for the casorati curvatures of submanifolds in generalized space forms endowed with semi-symmetric non-metric connections. Symmetry 2016, 8, 113. [Google Scholar] [CrossRef]
  26. Liaqat, M.; Laurian, P.; Othman, W.A.M.; Ali, A.; Gani, A.; Ozel, C. Estimation of inequalities for warped product semi-slant submanifolds of Kenmotsu space forms. J. Inequal. Appl. 2016, 2016, 239. [Google Scholar] [CrossRef]
  27. Li, J.; He, G.; Zhao, P. On Submanifolds in a Riemannian Manifold with a Semi-Symmetric Non-Metric Connection. Symmetry 2017, 9, 112. [Google Scholar] [CrossRef]
  28. Matsumoto, K.; Mihai, I. Warped product submanifolds in Sasakian space forms. SUT J. Math. 2002, 38, 135–144. [Google Scholar]
  29. Uddin, S.; Chi, A.Y.M. Warped product pseudo-slant submanifolds of nearly Kaehler manifolds. An. Stiintifice Univ. Ovidius Constanta 2011, 19, 195–204. [Google Scholar]
  30. Uddin, S.; Al-Solamy, F.R.; Khan, K.A. Geometry of warped product pseudo-slant submanifolds in nearly Kaehler manifolds. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat 2016, 3, 223–234. [Google Scholar]
  31. Zhang, P.; Zhang, L. Casorati inequalities for submanifolds in a Riemannian manifold of quasi-constant curvature with a semi-symmetric metric connection. Symmetry 2016, 8, 19. [Google Scholar] [CrossRef]
  32. Chen, B.-Y.; Uddin, S. Warped product pointwise bi-slant submanifolds of Kaehler manifolds. Publ. Math. Debrecen 2018, 92, 183–199. [Google Scholar] [CrossRef]
  33. Srivastava, S.K.; Sharma, A. Pointwise pseudo-slant warped product submanifolds in a Kaehler Manifold. Mediterr. J. Math. 2017, 14, 20. [Google Scholar] [CrossRef]
  34. Sahin, B. Warped product pointwise semi-slant submanifolds of Kaehler manifolds. Port. Math. 2013, 70, 252–268. [Google Scholar] [CrossRef]
  35. Chen, B.-Y.; Gray, O.J. Pointwise slant submanifolds in almost Hermitian manifolds. Turk. J. Math. 2012, 79, 630–640. [Google Scholar]
  36. Bishop, R.L.; Neil, B.O. Manifolds of negative curvature. Trans. Am. Math. Soc. 1969, 145, 1–9. [Google Scholar] [CrossRef]

Share and Cite

MDPI and ACS Style

Ali, A.; Alkhaldi, A.H. Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications. Symmetry 2019, 11, 200. https://doi.org/10.3390/sym11020200

AMA Style

Ali A, Alkhaldi AH. Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications. Symmetry. 2019; 11(2):200. https://doi.org/10.3390/sym11020200

Chicago/Turabian Style

Ali, Akram, and Ali H. Alkhaldi. 2019. "Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications" Symmetry 11, no. 2: 200. https://doi.org/10.3390/sym11020200

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop