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Semiinvariant warped product submanifolds of almost contact manifolds
Journal of Inequalities and Applications volume 2012, Article number: 127 (2012)
Abstract
In this article, we have obtained necessary and sufficient conditions in terms of canonical structure F on a semiinvariant submanifold of an almost contact manifold under which the submanifold reduced to semiinvariant warped product submanifold. Moreover, we have proved an inequality for squared norm of second fundamental form and finally, an estimate for the second fundamental form of a semiinvariant warped product submanifold in a generalized Sasakian space form is obtained, which extend the results of Chen, AlLuhaibi et al., and Hesigawa and Mihai in a more general setting.
2000 Mathematics Subject Classification: 53C25; 53C40; 53C42; 53D15.
1 Introduction
Bishop and O'Neil [1] introduced the notion of warped product manifolds. These manifolds are generalization of Riemannian product manifolds and occur naturally, e.g., surface of revolution is a warped product manifold. With regard to physical applications of these manifolds, one may realize that space time around a massive star or a black hole can be modeled on a warped product manifolds for instance and warped product manifolds are widely used in differential geometry, Physics and as well as in different branches of Engineering. Due to wide applications of warped product submanifolds, this becomes a fascinating and interesting topic for research, and many articles are available in literature (c.f., [2–4]). CRwarped product was introduced by Chen [5]. He studied warped products CRsubmanifolds in the setting of Kaehler manifolds and showed that there does not exist warped product CRsubmanifolds of the form M_{⊥}×_{ f } M_{ T } , therefore, he considered warped product CRsubmanifolds of the types M_{ T } ×_{ f } M_{⊥} and established a relationship between the warping function f and the squared norm of the second fundamental form of the CRwarped product submanifolds in Kaehler manifolds [5]. In the available literature, many geometers have studied warped products in the setting of almost contact metric manifolds (c.f., [6–8]). Hesigawa and Mihai [9] obtained the inequality for squared norm of the second fundamental form in term of the warping function for contact CRwarped product in Sasakian manifolds. Recently Atceken [4] studied contact CRwarped product submanifolds in Cosymplectic spaceforms and obtained an inequality for second fundamental form in terms of warping function. After reviewing the literature, we realized that there is very few studies on the warped product submanifold for almsot contact manifolds so it will be worthwhile to study the warped product submnifolds in the setting of almost contact metric manifold. Since generalized Sasakian space forms include all the classes of almost contact metric manifold, so we have obtained an inequality for squared norm of second fundamental form for semiinvariant warped product submanifolds in the setting of generalized Sasakian space form.
2 Preliminaries
Let \stackrel{\u0304}{M} be a (2n + 1)dimensional C^{∞}differentiable manifold endowed with the almost contact metric structure (ϕ, ξ, η, g), where ϕ is a tensor field of type (1, 1), ξ is a vector field, η is a 1form and g is a Riemannian metric on \stackrel{\u0304}{M}, all these tensor fields satisfying.
for any X,Y\in T\stackrel{\u0304}{M}. Here, T\stackrel{\u0304}{M} is the standard notation for the tangent bundle of \stackrel{\u0304}{M}. The twoform Φ denotes the fundamental twoform and is given by g(X, ϕY) = Φ(X, Y). The manifold \stackrel{\u0304}{M} is said to be contact if Φ = dη.
Most of the geometric properties of a Riemannian manifold depend on the curvature tensor R of a manifold. It is well known that the sectional curvatures of a manifold determine curvature tensor completely. A Riemannian manifold with constant sectional curvature c is known as real space form and its curvature tensor is given by
A Sasakian manifold with constant ϕsectional curvatures is a Sasakian space form and it has a specific form of its curvature tensor. Similar notion also holds for Kenmotsu and Cosymplectic space form. In order to generalize the notion in a common frame, Alegere et al. [10] introduced the notion of generalized Sasakian space form. In this connection, a generalized Sasakian space form is defined as follows.
Given an almost contact metric manifold \stackrel{\u0304}{M}\left(\varphi ,\xi ,\eta ,g\right), we say that \stackrel{\u0304}{M} is a generalized Sasakian space form if there exist three functions f_{1}, f_{2}, and f_{3} on \stackrel{\u0304}{M} such that, the curvature tensor R is given by
Let M be a submanifold of an almost contact metric manifold \stackrel{\u0304}{M}. Then we denote the induced metric on M by the same symbol g where as the induced connection on M by ∇. With these notation, Gauss and Weingarten formulae are written as
for each X, Y ∈ TM and N ∈ T^{⊥}M, where ∇^{⊥} denotes the induced connection on the normal bundle T^{⊥}M. h and A_{ N } are the second fundamental form and the shape operator of the immersion of M into \stackrel{\u0304}{M} and they are related as
For any X ∈ TM and N ∈ T^{⊥}M, we write
where TX and tN are the tangential components of ϕX and ϕN, respectively, where as FX and fN are the normal components of ϕX and ϕN, respectively.
The covariant derivative of the tensors T, F, t, and f are defined as
On the other hand, the covariant derivative of the second fundamental form h is defined as
for any X, Y, Z ∈ TM. Let \stackrel{\u0304}{R} and R be the curvature tensors of the connections \stackrel{\u0304}{\nabla} and ∇ on \stackrel{\u0304}{M} and M, respectively. Then the equations of Gauss and Coddazi are given by
A submanifold M of \stackrel{\u0304}{M} is said to be semiinvariant submanifold if there exist on M, a differentiable invariant distribution D such that its orthogonal complementary distribution D^{⊥} is antiinvariant, i.e., ϕD_{ x } ∈ T_{ x }M and \varphi {D}_{x}^{\perp}\subset {T}_{x}^{\perp}M for each x ∈ M. For a semiinvariant submanifold of an almost contact metric manifold \stackrel{\u0304}{M}, we have
The structure vector field ξ is tangential to submanifold M, otherwise the submanifold is simply antiinvariant.
where μ denotes the orthogonal complemantry distribution of ϕD^{⊥} and is an invariant normal subbundle of T^{⊥}M under ϕ.
The orthogonal projection on TM of a semiinvariant submanifold M of an almost contact metric manifold are denoted by P_{1} and P_{2}, i.e., for any X ∈ TM we have
where P_{1}X ∈ D, P_{2}X ∈ D^{⊥} and η(X)ξ ∈ 〈ξ〉. It follows immediately that
Moreover, for any X, Y ∈ TM if we denote by {\mathcal{P}}_{X}Y and {\mathcal{Q}}_{X}Y, the tangential and normal parts of \left({\stackrel{\u0304}{\nabla}}_{X}\varphi \right)Y, then we have
and on using Equations (2.5)(2.11), we obtain
Definition 2.1 A semiinvariant submanifold M of an almost contact metric manifold is semiinvariant product if the distributions are involutive and their leaves are totally geodesic in M.
Definition 2.2 Let (B, g_{ B } ) and (F, g_{ F } ) be two Riemannian manifolds with Riemannian metric g_{ B } and g_{ F } , respectively, and f be a positive differentiable function on B. The warped product of B and F is the Riemannian manifold (B × F, g), where
For a warped product manifold N_{1}×_{ f } N_{2}, we denote by D_{1} and D_{2} the distributions defined by the vectors tangent to the leaves and fibers, respectively. In other words, D_{1} is obtained by the tangent vectors of N_{1} via the horizontal lift and D_{2} is obtained by the tangent vectors of N_{2} via vertical lift. In case of semiinvariant warped product submanifolds D_{1} and D_{2} are replaced by D and D^{⊥}, respectively.
The warped product manifold (B × F, g) is denoted by B × _{ f } F. If X is the tangent vector field to M = B × _{ f } F at (p, q) then
Bishop and O'Neill [1] proved the following
Theorem 2.1 Let M = B× _{ f } F be warped product manifolds. If X, Y ∈ TB and V, W ∈ TF then

(i)
∇ _{ X }Y ∈ TB

(ii)
{\nabla}_{X}V={\nabla}_{V}X=\left(\frac{Xf}{f}\right)V,

(iii)
{\nabla}_{V}W=\frac{g\left(V,W\right)}{f}\nabla f.
∇f is the gradient of f and is defined as
for all X ∈ TM.
Corollary 2.1 On a warped product manifold M = N_{1} × _{ f } N_{2}, the following statements hold

(i)
N _{1} is totally geodesic in M

(ii)
N _{2} is totally umbilical in M.
Moreover,
and
for any X ∈ D_{1} and Z, W ∈ D_{2}, where nor(∇ _{ Z }W) denotes the component of ∇ _{ Z }W in D_{1} and ∇f denotes the gradient of f.
3 Semiinvariant warped product submanifolds
Chen [11] obtained various conditions under which a CRsubmanifolds reduces to a CRproduct. In particular, he proved that a CRsubmanifold of a Kaehler manifold is a CRproduct if and only if \stackrel{\u0304}{\nabla}P=0. Since, warped products are the generalization of Riemannian products by taking this point Khan et al. [8] proved a characterization of CRwarped product of a Kaehler manifold in terms of P and F after that AlLuhaibi et al. [7] find charactraziation of CRwarped of nearly Kaehler manifolds in terms of P and F. In this section, we have obtained necessary and sufficient condition in terms of F, for which a semiinvariant submanifolds is a semiinvariant warped product submanifold in the setting of almost contact metric manifolds.
Throughout, this section, we denote N_{ T } and N_{⊥} the invariant and antiinvariant submanifolds, respectively, of an almost contact metric manifold \stackrel{\u0304}{M}. Warped product semiinvariant and semiinvariant warped product submanifolds of an almost contact metric manifold \stackrel{\u0304}{M} are represented by N_{⊥} × _{ f } N_{ T } and N_{ T } ×_{ f } N_{⊥} and we take N_{ T } tangential to ξ.
In terms of canonical structure F, we have the following charectrization.
Theorem 3.1 A semiinvariant submanifold M with involutive distributions D ⊕ 〈ξ〉 and D^{⊥} of an almost contact metric manifold with {\mathcal{Q}}_{D}{D}^{\perp}\in \mu. Then M is a semiinvariant warped product submanifold of \stackrel{\u0304}{M} if and only if
for each U, V ∈ TM and μ a C^{∞} function on M satisfying Zμ = 0 for each Z ∈ D^{⊥}.
Proof. Let M be a semiinvariant warped product submanifold N_{ T } × _{ f } N_{⊥}, then, by Equation (2.11), we have
for any X, Y ∈ D and W ∈ D^{⊥}. As N_{ T } is totally geodesic in M, we get
On other hand, for any X ∈ D, Z, W ∈ D^{⊥} by Equation (2.23)
By the assumption that {\mathcal{Q}}_{D}{D}^{\perp}\in \mu, the above equation gives
As ξ is tangential to N_{ T } , for any X ∈ D and Z, W ∈ D^{⊥}, by Equations (2.11) and (2.25) we have
Similarly, for any Z, W, W' ∈ D^{⊥}, by Equation (2.11) we have
Moreover, for any X ∈ D, W ∈ D^{⊥} and ξ ∈ 〈ξ〉, by Equations (2.23) and (2.25), it is easy to see that
Since ξ is tangential to N_{ T } , then from Equations (2.11) and (2.25), we can prove the following
For any U, V ∈ TM with the help of Equation (2.19), we have
In view of Equations (3.2)(3.10), the above equation reduced to Equation (3.1).
Conversely, suppose that M be a semiinvariant submanifold, satisfying Equation (3.1). Then for any X, Y ∈ D ⊕ 〈ξ〉 by Equation (3.1), we have
therefore, by Equation (2.11) the above equation yields g(∇ _{ X }Y, W) = 0, this mean leaves of D ⊕ 〈ξ〉 are totally geodesic in M.
Now, for any Z, W ∈ D^{⊥}, by Equation (3.1), we get
or
Let N_{⊥} be a leaf of D_{⊥}. If ∇' denotes the induced connection on N_{⊥} and h' be the second fundamental form of the immersion of N_{⊥} of M, then by Gauss formula
or
or
this shows that N_{⊥} is totally umbilical in M with mean curvature vector ∇μ. Moreover, as Wμ = 0 for all W ∈ D^{⊥} and the mean curvature is parallel on N^{⊥}, this shows that N_{⊥} is extrinsic sphere. Hence, by virtue of result of [12] which says that if the tangent bundle of a Riemannian manifold M splits into an orthogonal sum TM = E_{0} ⊕ E_{1} of nontrivial vector subbundles such that E_{1} is spherical and its orthogonal complement E_{0} is auto parallel, then the, manifold M is locally isometric to a warped product M_{0} × _{ f } M_{1}, we can say M is a locally semiinvariant product submanifold N_{ T } ×_{ f } N_{⊥}, where warping function f = e^{μ} .
Let M = N_{ T } ×_{ f } N_{⊥} be a semiinvariant warped product submanifold of an almost contact metric manifold \stackrel{\u0304}{M}.
In view of decomposition (2.18), we may write
for each U, V ∈ TM, where {h}_{\varphi {D}^{\perp}}\left(U,V\right)\in \varphi {D}^{\perp} and h_{ μ } (U, V) ∈ μ.
If {e_{1}, e_{2},..., e_{ n } } be a local orthonormal frame of vector fields on M then we define
and for differentiable function f on M, the Laplacian Δf of f is defined as
Now we have the following proposition
Proposition 3.1 Let M be a semiinvariant warped product submanofold N_{ T } ×_{ f } N_{⊥} of an almost contact metric manifold of \stackrel{\u0304}{M}. Then

(i)
{h}_{\varphi {D}^{\perp}}\left(\varphi X,Z\right)=\left(X\mathsf{\text{ln}}f\right)\varphi Z+\varphi {\mathcal{P}}_{Z}\varphi X

(ii)
g\left({\mathcal{Q}}_{Z}X,\varphi W\right)=g\left({\mathcal{P}}_{Z}\varphi X,W\right)

(iii)
g\left(h\left(\varphi X,Z,\varphi h\left(X,Z\right)\right)\right.=\phantom{\rule{0.3em}{0ex}}{\u2225{h}_{\mu}\left(X,Z\right)\u2225}^{2}g\left({\mathcal{Q}}_{X}Z,\varphi {h}_{\mu}\left(X,Z\right)\right), for any X ∈ TN_{ T } and Z, W ∈ TN _{⊥}.
Proof. By Gauss formula
using the decomposition (2.21) and Equation (2.25), we get
Comparing tangential parts in above equation
taking inner product with W ∈ D^{⊥} on both side, we get
or equivalently
or
which proves the part (i) of proposition.
Now, on comparing the normal parts
or
taking inner product with ϕW and using Equation (3.17), we get
Taking inner product with ϕh(X, Z) in Equation (3.14), we find
which is the part (iii) of proposition.
For semiinvariant warped product submaniolds of an almost conatct metric manifold, we have the following theorem
Theorem 3.2 Let M = N_{ T } ×_{ f } N_{⊥} be a semiinvariant warped product submanifold of an almost contact manifold \stackrel{\u0304}{M} with {\mathcal{P}}_{{D}_{\perp}}D\in D, then

(i)
The squared norm of the second fundamental form satisfies
{\u2225h\u2225}^{2}\ge 2q{\u2225\nabla \mathsf{\text{ln}}f\u2225}^{2}+{\u2225{\mathcal{P}}_{{D}_{\perp}}D\u2225}^{2},(3.15)
where ∇ ln f is the gradient of ln f and q is the dimension of antiinvariant distribution.

(ii)
If the equality sign in (3.15) holds identically, then N_{ T } is totally geodesic submanifolds of \stackrel{\u0304}{M}, N _{⊥} is a totally umbilical submanifold of \stackrel{\u0304}{M}, M is minimal and h(D ⊕ D ^{⊥}, ξ) = 0.
Proof. Let {X_{0} = ξ, X_{1}, X_{2},..., X_{ p }, X_{p+1}= ϕX_{1},..., X_{2p}= ϕX_{ p } } be a local orthonormal frame of vector field on N_{ T } and {Z_{1}, Z_{2},..., Z_{ q } } be a local orthonormal frame of vector field on N_{⊥}. Then by definition of squared norm of mean curvature vector
Thus,
On using part (i) of Proposition (3.1) with assumption {\mathcal{P}}_{{D}_{\perp}}D\in D, then the above inequality takes the form
Using {\sum}_{i=1}^{2p}{\sum}_{r=1}^{q}{\u2225{\mathcal{P}}_{{Z}_{r}}{X}_{i}\u2225}^{2}=\phantom{\rule{2.77695pt}{0ex}}{\u2225{\mathcal{P}}_{{D}_{\perp}}D\u2225}^{2}, the above inequality can be represented as
which proves the part (i) of the Theorem.
Finally, if equality holds identically then from Equation (3.16), h(D, D) = 0, h(D^{⊥}, D^{⊥}) = 0, h(D, D^{⊥}) ⊆ ϕD^{⊥}, and h(D ⊕ D^{⊥}, ξ) = 0. These observations proves the part (ii) of theorem.
4 Semiinvariant warped product submanifolds of a generalized Sasakian spaceform
Hesigawa and Mihai [9] obtained the inequality for squared norm of second fundamental form for contact CRwarped product submanifolds in the setting of Sasakian space form. In the available literature, similar estimates are proved for squared norm of second fundamental form in contact manifolds (c.f., [3, 4]). Since generalized Sasakian space form include the class of all almost contact metric manifold, so in this section we will obtain an estimate for the squared norm of second fundamental form for semiinvariant warped product submanifolds in the setting of generalized Sasakian space form.
Theorem 4.1 Let M = N_{ T } ×_{ f } N_{⊥} be a semiinvariant warped product submanifold of a generalized Sasakian space form with {\mathcal{P}}_{{D}_{\perp}}D\in D. Then we have
Proof. For X ∈ D and Z ∈ D^{⊥}, by formula (2.4) we have
On the other hand by Coddazi equation
Now,
The first term in the righthand side of Equation (4.3) on using Equation (2.25), decomposition (2.25) and part (ii) of Proposition (3.1) becomes,
In view of assumption {\mathcal{P}}_{{D}^{\perp}}D\in D, the above equation gives
Where as, the second term of Equation (4.3) with the help of Equations (2.5) and (2.25) can be written as
By (i) and (ii) parts of Proposition 3.1, the above equation becomes
Applying Equation (3.14), (i) and (ii) parts of Proposition 3.1, we get
On substituting Equations (4.4) and (4.5) in Equation (4.3), we find
Similarly, we obtain
By formula (2.25) and part (i) of Proposition 3.1, we have
and
On using (i) and (ii) parts of Proposition 3.1 and the fact N_{ T } is totally geodesic, we have
and
The righthand side of above equation, on making use of the fact that N_{ T } is totally geodesic in M and the formula (2.25) reduced to g(∇ _{ Z }ϕ∇ _{ X }ϕX, Z), thus by using Gauss formula, we find
Let {X_{0} = ξ, X_{1}, X_{2},..., X_{ p }, X_{p+1}= ϕX_{1},..., X_{2p}= X_{ p } } and {Z_{1}, Z_{2},..., Z_{ q } } be a local orthonormal frame of vector fields on N_{ T } and N_{⊥}, respectively. Choosing X, Z as basic vector fields and substituting from Equations (4.3)(4.10) into Equation (4.2), we obtain
Summing both side over i = 1, 2,..., p, r = 1, 2,..., q and making use of Equation (4.1), we obtain
Here we use
Finally, on the same line of the proof of Equation (3.15), we can prove
The result follows immideatly from Equations (4.11) and (4.12).
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Acknowledgements
Authors are thankful to the referee for his valuable suggestion and comments. This work is supported by the Research Grant number 01361432S, Deanship of Scientific research, University of Tabuk, K.S.A.
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AlSolamy, F.R., Khan, M.A. Semiinvariant warped product submanifolds of almost contact manifolds. J Inequal Appl 2012, 127 (2012). https://doi.org/10.1186/1029242X2012127
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DOI: https://doi.org/10.1186/1029242X2012127
Keywords
 warped product
 semiinvariant
 almost contact metric manifold
 generalized Sasakian spaceform