In this article, we obtain the necessary and sufficient conditions that the semi-invariant submanifold to be a locally warped product submanifold of invariant and anti-invariant submanifolds of a cosymplectic manifold in terms of canonical structures T and F. The inequality and equality cases are also discussed for the squared norm of the second fundamental form in terms of the warping function.
The authors declare that they have no competing interests.
Authors' contributions
MAK carried out the geometric properties of the leaves of the involved distributions and participated to find out the geometric properties of warped products. SU participated in the study of warped products and drafted the manuscript. RS participated in the proof reading of the manuscript. All authors read and approved the final manuscript.
1 Introduction
Bishop and O'Neill [1] introduced the notion of warped product manifolds in order to construct a large variety of manifolds of negative curvature. Later on, the geometrical aspects of these manifolds have been studied by many researchers (c.f., [2‐5]). The idea of warped product submanifolds was introduced by Chen [6]. He studied warped product CR-submanifolds of the form M = M⊥ ×λMT such that M⊥ is a totally real submanifold and MT is a holomorphic submanifold of a Kaehler manifold and proved that warped product CR-submanifolds are simply CR-products. Therefore, he considered the warped product CR-submanifolds in the form of M = MT ×λM⊥ which are known as CR-warped products where MT and M⊥ are holomorphic and totally real submanifolds of a Kaehler manifold , respectively.
The warped product submanifolds of cosypmlectic manifolds was studied by Khan et.al [7]. Recently, Atçeken studied warped product CR-submanifolds of cosymplectic space form and obtained an inequality for the squared norm of the second fundamental form [2]. In this article, we obtain some basic results of semi-invariant submanifolds of cosymplectic manifolds and prove that a semi-invariant submanifold M of a cosymplectic manifold is locally a Riemannian product if and only if the canonical structure T is parallel. The semi-invariant warped product submanifolds are the generalization of locally Riemannian product submanifolds, so it will be worthwhile to study warped product submanifolds in terms of canonical structures T and F, to this end we obtain some characterization results on the warped product semi-invariant submanifolds in terms of the canonical structures T and F.
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2 Preliminaries
A (2m + 1)- dimensional C∞-manifold is said to have an almost contact structure if there exist on a tensor field ϕ of type (1, 1), a vector field ξ and 1-form η satisfying:
(2.1)
There always exists a Riemannian metric g on an almost contact manifold satisfying the following conditions
(2.2)
where X, Y are vector fields on .
An almost contact structure (ϕ, ξ, η) is said to be normal if the almost complex structure J on the product manifold is given by
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where f is the C∞ -function on has no torsion i.e., J is integrable. The condition for normality in terms of ϕ, ξ, and η is [ϕ, ϕ] + 2dη ⊗ ξ = 0 on , where [ϕ, ϕ] is the Nijenhuis tensor of ϕ. Finally, the fundamental two-form Φ is defined by Φ(X, Y) = g(X, ϕY).
An almost contact metric structure ( ϕ, ξ, η, g) is said to be cosymplectic, if it is normal and both Φ and η are closed [8], and the structure equation of a cosymplectic manifold is given by
(2.3)
for any X, Y tangent to , where denotes the Riemannian connection of the metric g on . Moreover, for cosymplectic manifold
(2.4)
Let M be a submanifold of an almost contact metric manifold with induced metric g and if ∇ and ∇⊥ are the induced connections on the tangent bundle TM and the normal bundle T⊥M of M, respectively. Denote by the algebra of smooth functions on M and by Γ(TM) the -module of smooth sections of a vector bundle TM over M, then the Gauss and Weingarten formulae are given by
(2.5)
(2.6)
for each X, Y ∈ Γ(TM) and V ∈ Γ(T⊥M), where h and AV are the second fundamental form and the shape operator (corresponding to the normal vector field V) respectively, for the immersion of M into . They are related by
(2.7)
where g denotes the Riemannian metric on as well as on M. The mean curvature vector H on M is given by
where n is the dimension of M and {e1, e2, . . . , en} is a local orthonormal frame of vector fields on M. The squared norm of the second fundamental form is defined as
For any X ∈ Γ(TM), we write
(2.8)
where TX and FX are the tangential and normal components of ϕX, respectively.
Similarly, for any V ∈ Γ(T⊥M), we write
(2.9)
where tV is the tangential component and fV is the normal component of ϕV. The covariant derivatives of the tensors T and F are defined as
(2.10)
(2.11)
for all X, Y ∈ Γ(TM).
Let M be a Riemannian manifold isometrically immersed in an almost contact metric manifold , then for every x ∈ M there exist a maximal invariant subspace denoted by Dx of the tangent space TxM of M. If the dimension of Dx is same for all values of x ∈ M, then Dx gives an invariant distribution D on M.
A submanifold M of an almost contact metric manifold is called a semi-invariant submanifold if there exist on M a differentiable distribution D whose orthogonal complementary distribution D⊥ is anti-invariant, i.e.,
(i)
TM = D ⊕ D⊥ ⊕ 〈ξ〉
(ii)
D is an invariant distribution
(iii)
D⊥ is an anti-invariant distribution i.e., ϕD⊥ ⊆ T⊥M.
A semi-invariant submanifold is anti-invariant if Dx = {0} and invariant if respectively, for every x ∈ M. It is a proper semi-invariant submanifold if neither Dx = {0} nor , for each x ∈ M.
Let M be a semi-invariant submanifold of an almost contact metric manifold . Then, FTxM is a subspace of such that
(2.12)
where ν is the invariant subspace of T⊥M under ϕ.
Let M be a proper semi-invariant submanifold of an almost contact metric manifold , then for any X ∈ Γ(TM), we have
(2.13)
where P1 and P2 are the orthogonal projections from TM to D and D⊥, respectively. It follows immediately that
(2.14)
From (2.3), (2.5), (2.6), (2.8), and (2.9), we have
(2.15)
(2.16)
for any X, Y ∈ Γ(TM).
Definition 2.1A semi-invariant submanifold M is said to be a locally semi-invariant product submanifold if M is locally a Riemannian product of the leaves of distributions D, D⊥, and 〈ξ〉.
Definition 2.2Let (N1, g1) and (N2, g2) be two Riemannian manifolds with Riemannian metrics g1and g2, respectively, and λ be a positive differentiable function on N1. Then the warped product of N1and N2is the Riemannian manifold (N1 × N2, g), where
The warped product manifold (N1 × N2, g) is denoted by N1 ×λ N2. If U is any vector field tangent to M = N1 ×λN2at (p, q), then
where π1and π2are the canonical projections of M onto N1and N2, respectively.
Bishop and O'Neill [1] proved the following results:
Theorem 2.1Let M = N1 ×λN2be a warped product manifold. If X, Y ∈ Γ(TN1) and Z, W ∈ Γ(TN2), then
(i)
∇XY ∈ Γ(TN1)
(ii)
(iii)
.
whereis the connection on N2and ∇λ is the gradient of the function λ and is defined as
(2.17)
for each U ∈ Γ(TM).
Corollary 2. 1On a warped product manifold M = N1 ×λN2, we have
(i)
N1is totally geodesic in M,
(ii)
N2is totally umbilical in M.
3 Some basic results on semi-invariant submanifolds
In the following section, we discuss some basic results on semi-invariant submanifolds of a cosymplectic manifold for later use. First, we obtain the integrability conditions of involved distributions in the definition of a semi-invariant submanifold and then we will see the geometric properties of their leaves.
Proposition 3.1 [9] Let M be a semi-invariant submanifold of a cosymplectic manifold then the anti-invariant distribution D⊥is integrable.
Proposition 3.2The invariant distribution D on a semi-invariant submanifold of a cosymplectic manifold is integrable if and only if
for each X, Y ∈ Γ(D) and Z ∈ Γ(D⊥).
Proof. The result can be obtained by making use of (2.2), (2.3), and (2.5). ■
Proposition 3.3If the invariant distribution D on a semi-invariant submanifold M of a cosymplectic manifoldis integrable, then its leaves are totally geodesic in M if and only if
for each U ∈ Γ(TM) and Y ∈ Γ(D).
Proof. From (2.16), we obtain
for any U ∈ Γ(TM) and Y ∈ Γ(D). Taking the inner product with ϕZ for any Z ∈ Γ(D⊥), we get
The result follows from the above equation. ■
Now, we have the following corollary for later use.
Corollary 3.1The invariant distribution D on a semi-invariant submanifold M of a cosymplectic manifoldis integrable and its leaves are totally geodesic in M if and only if
for any X, Y ∈ Γ(D).
Proof. The result follows from (2.15) and Proposition 3.3. ■
Lemma 3.1For a semi-invariant submanifold M of a cosymplectic manifold , the leaf N⊥of D⊥is totally geodesic in M if and only if
for any X ∈ Γ(D) and Z, W ∈ Γ(D⊥).
Proof. From (2.2), (2.3), (2.5), and (2.6), we obtain
Thus, the result follows from the above equation. ■
Theorem 3.1A semi-invariant submanifold M of a cosymplectic manifoldis locally a semi-invariant product if and only if
for any U, V ∈ Γ(TM).
Proof. If T is parallel then by (2.15), we have
(3.1)
for any U, V tangent to M. In particular, if X ∈ Γ(D), then (3.1) gives, th(U, X) = 0, that is,
(3.2)
for any Z ∈ Γ(D⊥). Thus by Proposition 3.2 and Lemma 3.1, D is integrable and the leaf N⊥ of D⊥ is totally geodesic in M. Let NT be a leaf of D, now for any X, Y ∈ Γ(D) and Z ∈ Γ(D⊥) by (3.2), we obtain g(AϕZX, Y) = 0 and using (2.2), (2.3), (2.5), and (2.6), we get g(∇XϕY, Z) = 0, which shows that leaf of D is totally geodesic in M and distribution 〈ξ〉 is already totally geodesic in M and hence M is locally a semi-invariant product.
Conversely, if M is locally a semi-invariant product then ∇U× ∈ Γ(D) for any X ∈ Γ(D) and U ∈ Γ(TM), thus by (2.15) and the Proposition 3.3, we get . Similarly, for any Z ∈ Γ(D⊥) and U ∈ Γ(TM), we obtain ∇UZ ∈ Γ(D⊥) and then by (2.10), we get and it is easy to see that . By these observations we find that , for all U, V ∈ Γ(TM), this proves the theorem completely. ■
4 Semi-invariant warped product submanifolds
Throughout this section, we denote NT and N⊥ the invariant and anti-invariant submanifolds of a cosymplectic manifold , respectively. The warped product semi-invariant submanifolds of a cosymplectic manifold are denoted by N⊥ ×λNT and NT ×λN⊥. The first type of warped products do not exist of a cosymplectic manifold in the sense of [5], here we discuss the second type of warped products and obtain some interesting results. First, we have the following lemma:
Lemma 4.1Let M = NT ×λN⊥be a warped product semi-invariant submanifold of an almost contact metric manifold . Then
for any X, Z, and U tangent to NT, N⊥, and M, respectively.
Proof. Let M = NT ×λN⊥ be a warped product submanifold of invariant and anti-invariant submanifolds of an almost contact metric manifold , then by Theorem 2.1 (ii), we have
(4.1)
for X ∈ Γ(TNT) and Z ∈ Γ(TN⊥). Then, from (2.10) and (4.1), we get
which proves the first part of the lemma. Now, for any U ∈ Γ(TM), we have TU ∈ Γ(TNT), therefore for any U ∈ Γ(TM). Furthermore, for any X ∈ Γ(TNT), we obtain
Using (2.13), the above equation reduced to
Using (4.1), the second term of right hand side is identically zero, then the above equation takes the form
Using (2.17), we obtain
That is,
This proves the lemma completely. ■
Theorem 4.1A proper semi-invariant submanifold of a cosymplectic manifoldis locally a warped product semi-invariant submanifold if and only if
(4.2)
for each U, V ∈ Γ(TM) and μ, a C∞ -function on M satisfying W μ = 0, for each W ∈ Γ(D⊥).
Proof. Let M = NT ×λN⊥ be a warped product semi-invariant submanifold of a cosymplectic manifold , then from (2.10) and (2.13), we have
Again using (2.10) and (2.13), the above equation takes the form
(4.3)
Now, from Lemma 4.1, we have
and
Substituting these values in (4.3), we obtain
Conversely, suppose that M is a semi-invariant submanifold of a cosymplectic manifold and (4.2) holds, then , for each X, Y ∈ Γ(D). Then by Corollary 3.1, D is integrable and each leave NT of D is totally geodesic in M. Moreover, from (4.2), we have
for X ∈ Γ(D) and Z, W ∈ Γ(D⊥). Using (2.3), (2.8), and (2.10), we obtain
That is,
Using cosymplectic character and (2.5), we derive
By (2.17), the above equation takes the form
(4.4)
Let us assume that N⊥ is a leaf of D⊥ and h' is the second fundamental form of the immersion of N⊥ into M, then
Using (4.4), we get
or,
This means that N⊥ is totally umbilical in M with non vanishing mean curvature ∇μ. Also, as W μ = 0, for all W ∈ Γ(D⊥), i.e., the mean curvature vector of N⊥ is parallel and the leaves of D⊥ are extrinsic spheres in M. Hence from a result of Hiepko [10], the submanifold M is locally a warped product semi-invariant submanifold of NT and N⊥ with warping function λ = eμ. ■
Note. Theorem 4.1 is a generalization of Theorem 3.1, and shows that what is the effect on , when the submanifold is a warped product semi-invariant submanifold.
Theorem 4.2A semi-invariant submanifold M of a cosymplectic manifoldis locally a warped product semi-invariant submanifold if and only if
(4.5)
for U, V ∈ Γ(TM) and W ∈ Γ(D⊥), where μ is a C∞ -function on M such that Z μ = 0, for all Z ∈ D⊥.
Proof. If M = NT ×λN⊥ is a warped product semi-invariant submanifold of a cosymplectic manifold , then NT and N⊥ are totally geodesic and totally umbilical in M, respectively. Moreover, we have
(4.5.1)
for any X ∈ Γ(D) and Z ∈ Γ(D⊥). Now, by (2.13), we have
Again, using (2.13), the above equation takes the form
In view of (2.4), (2.5), and (2.16), the above equation reduced to
Taking the inner product with ϕW, for any W ∈ Γ(D⊥), we obtain
Using (2.14), (2.16) and the fact that P1U ∈ Γ(D) and P2U ∈ Γ(D⊥), for any U ∈ Γ(TM), then the above equation becomes
From (2.2), the above equation becomes
Using (2.5), we derive
Using the covariant differentiation property of ϕ and the fact that P1V ∈ Γ(D) and P2V ∈ Γ(D⊥), for any V ∈ Γ(TM), then from (2.2), we obtain
Again using (2.5), we arrive at
The first term of right-hand side is zero by (4.1) and the fact that P1V ∈ Γ(D) and W ∈ Γ(D⊥), thus we obtain
Conversely, suppose that M is a semi-invariant submanifold of a cosymplectic manifold satisfying (4.5), then it is easy to see that
for each X, Y ∈ Γ(D) and W ∈ Γ(D⊥). Thus, by (2.16) we obtain
Therefore by Propositions 3.2 and 3.3, the distribution D is integrable and its leaves are totally geodesic in M. Now for any Z ∈ Γ(D⊥), by (4.5), we have
Using (2.16), we get
(4.6)
Let N⊥ be a leaf of D⊥ and h' be the second fundamental form of the immersion of N⊥ into M and ∇' is the induced connection on N⊥, then by Gauss formula, we have
(4.7)
Now for any Z, W ∈ Γ(D⊥) and X ∈ Γ(D), by (2.3) and (2.5), we have
From (4.7), we obtain
(4.8)
Thus, by (4.6) and (4.8), we derive
Using (2.17), we obtain
which implies that N⊥ is totally umbilical in M with non vanishing mean curvature vector ∇μ. Moreover, as Z μ = 0 for all Z ∈ Γ(D⊥) that is, the mean curvature is parallel on N⊥, this show that N⊥ is extrinsic sphere. Hence, from a result of [10], M is locally a warped product submanifold. ■
Proposition 4.1. Let M = NT ×λN⊥be a warped product semi-invariant submanifold of a cosymplectic manifold of . Then
(i)
(ii)
g(h(ϕX, Z), ϕh(X, Z)) = ||hν(X, Z)||2
for any × ∈ Γ(D) and Z ∈ Γ(D⊥).
Proof. For any X ∈ Γ(D) and Z ∈ Γ(D⊥), by Gauss formula, we have
Using (4.1), we get
(4.9)
Equating the tangential components of (4.9), we get
Taking the inner product with W ∈ Γ(D⊥), we obtain
or equivalently
Replacing X by ϕX, we obtain
which proves the part (i) of proposition. Now, for the second part comparing the normal components of (4.9), we get
or,
Taking the inner product with ϕ h(X, Z), we derive
which completes the proof. ■
Theorem 4.3. Let M = NT ×λN⊥be a warped product semi-invariant submanifold of a cosymplectic manifold . Then
(i)
The squared norm of the second fundamental form satisfies
where ∇ ln λ is the gradient of the function ln λ and q is the dimension of N⊥.
(ii)
If the equality holds identically, then NTis a totally geodesic submanifold of, N⊥is a totally umbilical submanifold of and M is minimal.
Proof. Let {X1, X2, . . . , Xp, Xp+1= ϕX1, . . . , X2p= ϕXp, X2p+1= ξ} be a local orthonormal frame of vector fields on NT and {Z1, Z2, . . . , Zq} a local orthonormal frame on N⊥. Then by definition of squared norm of mean curvature vector
(4.10)
or,
In view of Proposition 4.1 (i), we get
This verifies the assertion (i). If the equality sign holds, then from (4.10) and Proposition 4.1 (i), we get
(4.11)
As NT is a totally geodesic submanifold of M, the first condition of (4.11) implies that NT is totally geodesic in . Moreover, N⊥ is totally umbilical in M, the second condition of (4.11) implies that N⊥ is totally umbilical in , and also it follows from (4.11) that M is minimal in . ■
Acknowledgements
MAK was supported by the Research Grant 0136-1432-S, Deanship of Scientific research, (University of Tabuk, K.S.A.) and SU was supported by the Grant RG117/10AFR (University of Malaya, Malaysia).
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
MAK carried out the geometric properties of the leaves of the involved distributions and participated to find out the geometric properties of warped products. SU participated in the study of warped products and drafted the manuscript. RS participated in the proof reading of the manuscript. All authors read and approved the final manuscript.