Semi-invariant warped product submanifolds of almost contact manifolds
© Al-Solamy and Khan; licensee Springer. 2012
Received: 1 December 2011
Accepted: 7 June 2012
Published: 7 June 2012
In this article, we have obtained necessary and sufficient conditions in terms of canonical structure F on a semi-invariant submanifold of an almost contact manifold under which the submanifold reduced to semi-invariant 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 semi-invariant warped product submanifold in a generalized Sasakian space form is obtained, which extend the results of Chen, Al-Luhaibi et al., and Hesigawa and Mihai in a more general setting.
2000 Mathematics Subject Classification: 53C25; 53C40; 53C42; 53D15.
Bishop and O'Neil  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]). CR-warped product was introduced by Chen . He studied warped products CR-submanifolds in the setting of Kaehler manifolds and showed that there does not exist warped product CR-submanifolds of the form M⊥× f M T , therefore, he considered warped product CR-submanifolds 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 CR-warped product submanifolds in Kaehler manifolds . 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  obtained the inequality for squared norm of the second fundamental form in term of the warping function for contact CR-warped product in Sasakian manifolds. Recently Atceken  studied contact CR-warped product submanifolds in Cosymplectic space-forms 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 semi-invariant warped product submanifolds in the setting of generalized Sasakian space form.
for any . Here, is the standard notation for the tangent bundle of . The two-form Φ denotes the fundamental two-form and is given by g(X, ϕY) = Φ(X, Y). The manifold is said to be contact if Φ = dη.
A Sasakian manifold with constant ϕ-sectional 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.  introduced the notion of generalized Sasakian space form. In this connection, a generalized Sasakian space form is defined as follows.
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.
where μ denotes the orthogonal complemantry distribution of ϕD⊥ and is an invariant normal subbundle of T⊥M under ϕ.
Definition 2.1 A semi-invariant submanifold M of an almost contact metric manifold is semi-invariant product if the distributions are involutive and their leaves are totally geodesic in M.
For a warped product manifold N1× f N2, we denote by D1 and D2 the distributions defined by the vectors tangent to the leaves and fibers, respectively. In other words, D1 is obtained by the tangent vectors of N1 via the horizontal lift and D2 is obtained by the tangent vectors of N2 via vertical lift. In case of semi-invariant warped product submanifolds D1 and D2 are replaced by D and D⊥, respectively.
Bishop and O'Neill  proved the following
∇ X Y ∈ TB
for all X ∈ TM.
N 1 is totally geodesic in M
N 2 is totally umbilical in M.
for any X ∈ D1 and Z, W ∈ D2, where nor(∇ Z W) denotes the component of ∇ Z W in D1 and ∇f denotes the gradient of f.
3 Semi-invariant warped product submanifolds
Chen  obtained various conditions under which a CR-submanifolds reduces to a CR-product. In particular, he proved that a CR-submanifold of a Kaehler manifold is a CR-product if and only if . Since, warped products are the generalization of Riemannian products by taking this point Khan et al.  proved a characterization of CR-warped product of a Kaehler manifold in terms of P and F after that Al-Luhaibi et al.  find charactraziation of CR-warped 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 semi-invariant submanifolds is a semi-invariant warped product submanifold in the setting of almost contact metric manifolds.
Throughout, this section, we denote N T and N⊥ the invariant and anti-invariant submanifolds, respectively, of an almost contact metric manifold . Warped product semi-invariant and semi-invariant warped product submanifolds of an almost contact metric manifold 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.
for each U, V ∈ TM and μ a C ∞ function on M satisfying Zμ = 0 for each Z ∈ D⊥.
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.
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  which says that if the tangent bundle of a Riemannian manifold M splits into an orthogonal sum TM = E0 ⊕ E1 of nontrivial vector subbundles such that E1 is spherical and its orthogonal complement E0 is auto parallel, then the, manifold M is locally isometric to a warped product M0 × f M1, we can say M is a locally semi-invariant product submanifold N T × f N⊥, where warping function f = e μ .
Let M = N T × f N⊥ be a semi-invariant warped product submanifold of an almost contact metric manifold .
for each U, V ∈ TM, where and h μ (U, V) ∈ μ.
Now we have the following proposition
, for any X ∈ TN T and Z, W ∈ TN ⊥.
which proves the part (i) of proposition.
which is the part (iii) of proposition.
For semi-invariant warped product submaniolds of an almost conatct metric manifold, we have the following theorem
- (i)The squared norm of the second fundamental form satisfies(3.15)
If the equality sign in (3.15) holds identically, then N T is totally geodesic submanifolds of , N ⊥ is a totally umbilical submanifold of , M is minimal and h(D ⊕ D ⊥, ξ) = 0.
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 Semi-invariant warped product submanifolds of a generalized Sasakian space-form
Hesigawa and Mihai  obtained the inequality for squared norm of second fundamental form for contact CR-warped 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 semi-invariant warped product submani-folds in the setting of generalized Sasakian space form.
Authors are thankful to the referee for his valuable suggestion and comments. This work is supported by the Research Grant number 0136-1432-S, Deanship of Scientific research, University of Tabuk, K.S.A.
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