Hemi-slant warped product submanifolds of nearly Kaehler manifolds are studied and some interesting results are obtained. Moreover, an inequality is established for squared norm of second fundamental form and equality case is also discussed. The results obtained are also true if ambient manifold is replaced by a Kaehler manifold. These results generalize several known results in the literature.

1. Introduction

In [1] Bishop and O’Neill introduced the notion of warped product manifolds as a natural generalization of Riemannian product manifolds. For instance, a surface of revolution is a warped product manifold. So far as its applications are concerned, it has been shown that warped product manifolds provide excellent setting to model space time near black holes or bodies with large gravitational forces (see [1, 2]). Due to wide applications of warped product submanifolds, this becomes a fascinating and interesting topic for research and many articles are available in literature (see [1, 3–5]). Chen [6] initiated the study of warped product submanifolds by showing that there do not exist warped product CR-submanifolds of the type N⊥×fNT, and he considered warped product CR-submanifolds of the types NT×fN⊥ and established a relationship between the warping function f and the squared norm of the second fundamental form. Extending the study of Chen, Sahin [7] proved that there exist no semislant warped product submanifolds in a Kaehler manifold. In [8], V. A. Khan and K. A. Khan studied generic warped product submanifolds of nearly Kaehler manifolds and obtained an inequality for squared norm of second fundamental form in terms of warping function. Recently, Sahin [4] investigated hemi-slant warped product for Kaehler manifolds and obtained an inequality for squared norm of second fundamental form for mixed totally geodesic submanifolds. In view of the interesting geometric characteristic of nearly Kaehler manifolds and the nonexistence of CR-product submanifolds in S6 [9], it will be significant to explore hemi-slant warped product submanifolds of a nearly Kaehler manifold. In this continuation we have achieved success in extending the results of Sahin [4] and Chen [6] to the setting of nearly Kaehler manifolds.

2. Preliminaries

Let (M-,J,g) be a nearly Kaehler manifold with an almost complex structure J and Hermitian metric g and a Levi-Civita connection ∇- such that
(1)g(JX,JY)=g(X,Y),(2)(∇-XJ)Y+(∇-YJ)X=0,
for all vector fields X and Y on M-. Six-dimensional sphere S6 is a classic example of a nearly Kaehler non-Kaehler manifold. It has an almost complex structure J defined by the vector cross product in the space of purely imaginary Cayley numbers which satisfies the condition (∇-XJ)X=0. Let C be the Cayley division algebra generated by {e0=1,ei, (1≤i≤7)} over R and C+ the subspace of C consisting of all purely imaginary Cayley numbers. We may identify C+ with a 7-dimensional Euclidean space R7 with the canonical inner product g=(,). The automorphism group of C is the compact simple Lie group G2 and furthermore the inner product g is invariant under the action of G2 and hence, the group G2 may be considered as a subgroup of SO(7). A vector cross product for vectors in R7(=C+) is defined by
(3)x×y=(x,y)e0+xy,∀x,y∈C+.
Then the multiplication table for ej×ek is given by
(4)jk123456710e3-e2e5-e4e7-e62-e30e1e6-e7-e4e53e2-e10-e7-e6e5e44-e5-e6e70e1e2-e35e4e7e6-e10-e3-e26-e7e4-e5-e2e30e17e6-e5-e4e3e2-e10.
Considering S6 as {x∈C+:(x,x)=1}, the almost complex structure J on S6 is defined by
(5)Jx(X)=x×X,
where x∈S6 and X∈TxS6. The almost complex structure defined by the above equation together with the induced metric on S6 from g on R7(=C+) gives rise to a nearly Kaehler structure S6 [10].

Let M be a submanifold of M-. Then the induced Riemannian metric on M is denoted by the same symbol g and the induced connection on M is denoted by the symbol ∇. If TM- and TM denote the tangent bundle on M- and M, respectively, and T⊥M, the normal bundle on M, then the Gauss and Weingarten formulae are given by
(6)∇-XY=∇XY+h(X,Y),(7)∇-XN=-ANX+∇X⊥N,
for X, Y∈TM and N∈T⊥M where ∇⊥ denotes the connection on the normal bundle T⊥M. h and AN are the second fundamental form and the shape operator of immersions of M into M-. Corresponding to the normal vector field V they are related as
(8)g(AVX,Y)=g(h(X,Y),V).
The mean curvature vector H of M is given by
(9)H=1n∑i=1nh(ei,ei),
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
(10)∥h∥2=∑i,j=1ng(h(ei,ej),h(ei,ej)).
A submanifold M of M- is said to be a totally geodesic submanifold if h(X,Y)=0 for each X, Y∈TM, and totally umbilical submanifold if h(X,Y)=g(X,Y)H.

For X∈TM and N∈T⊥M we write
(11)JX=PX+FX,JN=tN+fN,
where PX and tN are tangential components of JX and JN, respectively, and FX and fN are the normal components of JX and JN.

The covariant differentiation of the tensors P, F, t, and f is defined, respectively, as
(12)(∇-XP)Y=∇XPY-P∇XY,(13)(∇-XF)Y=∇X⊥FY-F∇XY,(14)(∇-Xt)N=∇XtN-t∇X⊥N,(15)(∇-Xf)N=∇X⊥fN-f∇X⊥N.
Furthermore, for any X, Y∈TM, the tangential and normal parts of (∇-XJ)Y are denoted by 𝒫XY and 𝒬XY; that is,
(16)(∇-XJ)Y=𝒫XY+𝒬XY.
On using (6)–(13), we may obtain that
(17)𝒫XY=(∇-XP)Y-AFYX-th(X,Y),(18)𝒬XY=(∇-XF)Y+h(X,TY)-fh(X,Y).
Similarly, for N∈T⊥M, denoting by 𝒫XN and 𝒬XN, respectively, the tangential and normal parts of (∇-XJ)N, we find that
(19)𝒫XN=(∇-Xt)N+PANX-AfNX,𝒬XN=(∇-Xf)N+h(tN,X)+FANX.
On a submanifold M of a nearly Kaehler manifold by (2) and (16)
(20)(a)𝒫XY+𝒫YX=0,(b)𝒬XY+𝒬YX=0
for any X, Y∈TM.

Let M- be an almost Hermitian manifold with an almost complex structure J and Hermitian metric g and let M be a submanifold of M-. Submanifold M is said to be CR-submanifold if there exist two orthogonal complementary distributions D and D⊥ such that D is holomorphic distribution, that is, JD⊆D and D⊥ is totally real distribution, that is, JD⊥⊆T⊥M.

An immersed submanifold M of an almost Hermitian manifold M- is said to be slant submanifold if the Wirtinger angle θ(X)∈[0,π/2] between JX and TxM, and X≠0 (cf. [11]). Holomorphic and totally real submanifolds are slant submanifolds with wirtinger angle 0 and π/2. A submanifold is called proper slant if it is neither holomorphic nor totally real. More generally, a distribution Dθ on M- is called a slant distribution if the angle θ(X) between JX and Dθ has the same value θ for each x∈M- and X∈Dθ and X≠0.

If M is a slant submanifold of an almost Hermitian manifold M-, then we have (cf. [11])
(21)P2X=-cos2θX,
where θ is the wirtinger angle of M in M-. Hence we have
(22)g(PX,PY)=cos2θg(X,Y),(23)g(FX,FY)=sin2θg(X,Y),
for any X, Y∈TM.

Papaghiuc [12] introduced a class of submanifolds in almost Hermitian manifolds called the semislant submanifolds; this class includes the class of proper CR-submanifolds and slant submanifolds. Cabrerizo et al. [13] initiated the study of contact version of semislant submanifold and also bislant submanifolds. As a step forward, Carriazo [14] defined and studied bislant submanifolds and simultaneously gave the notion of antislant submanifolds in almost Hermitian manifolds; after that V. A. Khan and M. A. Khan [15] studied antislant submanifolds with the name pseudo-slant submanifolds in the setting of Sasakian manifolds.

Recently, Sahin [4] renamed pseudo-slant submanifolds as hemi-slant submanifolds and studied hemi-slant submanifolds for their warped product.

Definition 1.

A submanifold M of an almost Hermitian manifold is called hemi-slant submanifold if it is endowed with two orthogonal complementary distributions D⊥ and Dθ such that D⊥ is totally real, that is, JD⊥⊆T⊥M and Dθ is slant distribution with slant angle θ.

It is straight forward to see that CR-submanifolds and slant submanifolds are hemi-slant submanifolds with θ=0 and D⊥={0}, respectively.

If μ is the invariant subspace of the normal bundle T⊥M then in the case of hemi-slant submanifold, the normal bundle T⊥M can be decomposed as follows:
(24)T⊥M=μ⊕FD⊥⊕FDθ.

As D⊥ and Dθ are orthogonal distributions on M, then it is easy to see that the distributions FD⊥ and FDθ are mutually perpendicular. In fact, the decomposition (24) is an orthogonal direct decomposition.

A hemi-slant submanifold M is called a hemi-slant product if the distributions D⊥ and Dθ are parallel on M. In this case M is foliated by the leaves of these distributions. In particular if M is CR-submanifold with parallel distribution then it is called CR-product. In general, if M1 and M2 are Riemannian manifolds with Riemannian metrics g1 and g2, respectively, then the product manifold (M1×M2, g) is a Riemannian manifold with Riemannian metric g defined as
(25)g(X,Y)=g1(dπ1X,dπ1Y)+g2(dπ2X,dπ2Y),
where πi(i=1,2) are the projection maps of M onto M1 and M2, respectively, and dπi(i=1,2) are their differentials.

As a generalization of the product manifold and in particular of a hemi-slant product submanifold, one can consider warped product of manifolds which are defined in the following.

Definition 2.

Let (B,gB) and (C,gC) be two Riemannian manifolds with Riemannian metrics gB and gC, respectively, and f a positive differentiable function on B. The warped product of B and C is the Riemannian manifold (B×C, g), where
(26)g=gB+f2gC.

For a warped product manifold N1×fN2, 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.

The warped product manifold (B×C, g) is denoted by B×fC. If X is the tangent vector field to M=B×fC at (p,q) then
(27)∥X∥2=∥dπ1X∥2+f2(p)∥dπ2X∥2.

Bishop and O’Neill [1] proved the following.

Theorem 3.

Let M=B×fC be warped product manifolds. If X, Y∈TB and V, W∈TC then

∇XY∈TB,

∇XV=∇VX=(Xf/f)V,

∇VW=(-g(V,W)/f)∇f.

∇f is the gradient of f and is defined as
(28)g(∇f,X)=Xf,
for all X∈TM.
Corollary 4.

On a warped product manifold M=N1×fN2, the following statements hold:

N1 is totally geodesic in M

N2 is totally umbilical in M.

In what follows, N⊥ and Nθ will denote a totally real and slant submanifold, respectively, of an almost Hermitian manifold M-.

A warped product manifold is said to be trivial if its warping function f is constant. More generally, a trivial warped product manifold M=N1×N2 is a Riemannian product N1×N2f, where N2f is the manifold with the Riemannian metric f2g2 which is homothetic to the original metric g2 of N2. For example, a trivial CR-warped product is CR-product.

Sahin [4] extended the study of warped product hemi-slant submanifolds and hemi-slant warped product of Kaehler manifolds introducing warped product submanifolds as Nθ×f2N⊥ and N⊥×fNθ, where θ is the slant angle.

3. Hemi-Slant Warped Product Submanifolds

In [5] Uddin and Chi investigated warped product pseudo-slant (hemi-slant) submanifolds of nearly Kaehler manifolds and they only showed that there do not exist warped products of the form N⊥×fNθ in nearly Kaehler manifolds, where N⊥ is totally real submanifold and Nθ is slant submanifold. In this section we study the warped products of the types Nθ×fN⊥.

Let M=Nθ×fN⊥ be a hemi-slant warped product of a nearly Kaehler manifold M-. Then by Theorem 3,
(29)∇XZ=∇ZX=XlnfZ,
for any X∈TNθ, Z∈TN⊥.

Now by formula (12) and Theorem 3,
(30)(∇-ZP)W=g(Z,W)P(∇lnf)-g(Z,PW)∇lnf,
for each Z, W∈TNθ. Now we will investigate some interesting results of the second fundamental form.

Proposition 5.

On a hemi-slant warped product submanifold M=Nθ×fN⊥ of a nearly Kaehler manifold M-, one has

2g(h(X,Y),FZ)=g(h(X,Z),FY)+g(h(Y,Z),FX),

g(h(X,Z),FX)=g(h(X,X),FZ),

for any X, Y∈Nθ and Z∈N⊥.
Proof.

As Nθ is totally geodesic in M then (∇-XP)Y∈TNθ and therefore by formula (17),
(31)g(𝒫XY,Z)=-g(AFXY,Z)-g(th(X,Y),Z),
or
(32)g(h(X,Y),FZ)=g(𝒫XY,Z)+g(h(Y,Z),FX).
Similarly, we have
(33)g(h(X,Y),FZ)=g(𝒫YX,Z)+g(h(X,Z),FY).
Adding above two equations and using (20)(a), we get part (i).

Now by formula (17) and (20)(a),
(34)0=(∇-XP)Z+(∇-ZP)X-2th(X,Z)-AFZX-AFXZ,
and by (29) the above equation gives
(35)(PXlnf)Z=2th(X,Z)+AFZX+AFXZ.
Taking inner product of (35) with X∈Dθ, we get
(36)g(h(X,Z),FX)=g(h(X,X),FZ),
which proves part (ii).

Theorem 6.

For a hemi-slant warped product submanifold Nθ×fN⊥ of a nearly Kaehler manifold the warping function satisfies the following relation:
(37)cos2θXlnf∥Z∥2=g(h(PX,Z),FZ)-g(H,FPX)∥Z∥2,
for any X∈TNθ and Z∈TN⊥.

Proof.

If M is a hemi-slant warped product submanifold Nθ×fN⊥ of a nearly Kaehler manifold then (∇-XP)Z=0 for each X∈TNθ and Z∈TN⊥, and thus by (17),
(38)𝒫XZ=-AFZX-th(X,Z).
On the other hand
(39)𝒫ZX=∇ZPX-P∇ZX-AFXZ-th(X,Z).
Now using (29), the above equation takes the form
(40)𝒫ZX=PXlnfZ-AFXZ-th(X,Z).
Adding (38) and (40) and using (20)(a),
(41)AFZX+AFXZ=PXlnfZ-2th(X,Z),
taking inner product with Z∈TN⊥, and using the fact that N⊥ is totally umbilical, one gets the following equation:
(42)PXlnf∥Z∥2=-g(h(X,Z),FZ)+g(H,FX)∥Z∥2.
By replacing X by PX the required result follows.

Remark 7.

In [4] Sahin proved that hemi-slant warped products of the type N⊥×fNθ do not exist in the setting of Kaehler manifolds. Therefore, in the following Corollary we discuss the warped products of the type Nθ×fN⊥.

Corollary 8.

For a hemi-slant warped product submanifold Nθ×fN⊥ of a Kaehler manifold the warping function satisfies the following relation:
(43)cos2θXlnf∥Z∥2=g(h(PX,Z),FZ)-g(H,FPX)∥Z∥2,
for any X∈TNθ and Z∈TN⊥.

Proof.

Since M is a hemi-slant warped product submanifold of a Kaehler manifold, then by tensorial equation of Kaehler manifold, it is easy to see that 𝒫ZX=0, for any X∈TNθ and Z∈TN⊥, and using this fact in (40) and taking inner product with Z∈TN⊥, we get the required result.

Let us denote by Dθ and D⊥ the tangent bundles on Nθ and N⊥, respectively, and let {X1,X2,…,Xp,Xp+1=PX1,…,X2p=PXp} and {Z1,Z2,…,Zq} be local orthonormal frames of vector fields on Nθ and N⊥, respectively, with 2p and q being their real dimensions; then
(44)∥h∥2=∑i,j=12pg(h(Xi,Xj),h(Xi,Xj))+∑i=12p∑r=1qg(h(Xi,Zr),h(Xi,Zr))+∑r,s=1qg(h(Zr,Zs),h(Zr,Zs)).

Now we calculate the inequality for the squared norm of second fundamental form in the following theorem.

Theorem 9.

Let M=Nθ×fN⊥ be a hemi-slant warped product of a nearly Kaehler manifold M- with Nθ and N⊥ slant and totally real submanifolds, respectively, of M-. If H∈μ, then

the second fundamental form satisfies the following inequality:
(45)∥h∥2≥4qcos2θ∥∇lnf∥2,

equality holds if h(D,D)=0, h(D⊥,D⊥)=0, h(PXi,Zr) is normal to FZt and h(PXi,Zr) is normal to FXi and FXs where i,s=1,2,…,2p and r,t=1,…,q (i≠s and r≠t),

where ∇lnf is the gradient of lnf and q is dimension of N⊥.
Proof.

In view of (24) the second fundamental form can be decomposed as follows:
(46)h(U,V)=hFDθ(U,V)+hFD⊥(U,V)+hμ(U,V),
for each U,V∈TM, where hFDθ(U,V)∈FDθ, hFD⊥(U,V)∈FD⊥ and hμ(U,V)∈μ with
(47)hFDθ(U,V)=∑i=12phi(U,V)FXi,
where,
(48)hi(U,V)=csc2θg(h(U,V),FXi),(49)hFD⊥(U,V)=∑r=1qhr(U,V)FZr,
where,
(50)hr(U,V)=g(h(U,V),FZr),
for each Xi∈Dθ and Zr∈D⊥.

Now, making use of (37) with assumption H∈μ and formulae (48) and (49), we obtain
(51)g(hFD⊥(PXi,Zr),hFD⊥(PXi,Zr))=cos2θhr(PXi,Zr)(Xilnf)+∑t≠r(ht(PXi,Zr))2.
Summing over i=1,…,2p and r=1,…,q and (37) and assumption H∈μ with formula (50) the above equation gives
(52)g(hFD⊥(PXi,Zr),hFD⊥(PXi,Zr))=2qcos4θ∥∇lnf∥2+∑i=12p∑r,t=1,r≠tq(ht(PXi,Zr))2.
Let us consider the orthonormal frame of vector fields on D⊥ as {Z1,…,Zq}, and the second term in the right hand side of last equation on using (50) can be written as
(53)∑i=12p[∑r=1q(g(h(PXi,Zr),FZr))2fffff+∑r=1q∑t=1,t≠rq(g(h(PXi,Zr),FZt))2].
On applying (37), the first part of above expression reduced to
(54)∑i=12p∑r=1q(g(h(PXi,Zr),FZr))2=2qcos4θ∥∇lnf∥2.
Taking account of the above equation into (52), we obtain
(55)g(hFD⊥(PXi,Zr),hFD⊥(PXi,Zr))≥4qcos4θ∥∇lnf∥2.
The inequality follows from (44) and (55).

To discuss the equality case we will explore the expression ∥hFDθ(PXi,Zr)∥2 as follows.

Making use of (47), (48), and (23) and summing over r=1,…,q and i=1,…,2p we find
(56)g(hFDθ(PXi,Zr),hFDθ(PXi,Zr))=sin2θ∑r=1q[{∑i=12p(hi(PXi,Zr))2sssssssssssssssissss+∑i,s=1,s≠i2p(hs(PXi,Zr))2}].
Let us choose the orthonormal frame of vectors fields on Dθ as {X1,…,Xp,PX1,PX2,…,PXp=X2p}. Then the right hand side of the above equation with the help of (48) can be written as
(57)g(hFDθ(PXi,Zr),hFDθ(PXi,Zr))=csc2θ∑r=1q[2∑i=12p(g(h(PXi,Zr),FXi))2ffffffffsiff+∑i=12p∑s=1,s≠i2p(g(h(PXi,Zr),FXs))2].
From (44), (50), and expression (53), it is clear that equality holds if h(D,D)=0, h(D⊥,D⊥)=0,h(PXi,Zr) is normal to FZt and h(PXi,Zr) is normal to FXi and FXs, i,s=1,2,…,2p, r,t=1,…,q, where i≠s and r≠t.

Remark 10.

Since (37) is the key result of the paper which helps to get the inequality in Theorem 9 and moreover (37) is also true for the Kaehler manifolds, hence the results in Theorem 9 are also true for hemi-slant warped product submanifolds of a Kaehler manifold.

Now we compile some results of [16] and give the following example of a warped product submanifold in S6.

Example 11.

Let {e0,ei(1≤i≤7)} be the canonical basis of the Cayley division algebra on R8 over R and R7 the subspace of R8 generated by the purely imaginary Cayley numbers ei(1≤i≤7). Then
(58)S6={x1e1+x2e2+⋯+x7e7:x12+x22+⋯+x72=1}⊂R7
is a unit sphere admitting a nearly Kaehler structure (J,g,∇-) as has been specified earlier. Now suppose that S2={x=(x2,x4,x6)∈R3:x22+x42+x62=1} is a unit 2-sphere. For a real triple p=(p1,p2,p3) with p1+p2+p3=0 and p1p2p3≠0, define a mapping ψp of S2×R to S5⊂S6 as follows:
(59)ψp(x1,x2,x3,t)=x1(cos(tp1)e1+sin(tp1)e5)+x-2(cos(tp2)e2+sin(tp2)e6)+x3(cos(tp3)e3+sin(tp3)e7),
where x12+x22+x32=1 and t∈R. Then ψp gives rise to an isometric immersion from warped product Riemannian manifold S2×fR into S6. Moreover, induced warped product metric g- on S2×fR is given by
(60)g-=π1*g0+(∑i=13(xipi)2)π2*dt2,
where π1:S2×fR→S2 and π2:S2×R→R are natural projections and g0 is the Riemannian metric on S2 and from (27) it is evident that warping function f is as follows:
(61)f=∑i=13(xipi)2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are highly thankful to anonymous referees for their valuable suggestions and comments which have improved the paper.

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