We study totally umbilical hemi-slant submanifolds of a Kaehler manifold via curvature tensor. We prove some classification theorems for totally umbilical hemi-slant submanifolds of a Kaehler manifold and give an example.

1. Introduction

The notion of slant submanifolds of an almost Hermitian manifold was introduced by Chen [1]. These submanifolds are the generalization of both holomorphic and totally real submanifolds of an almost Hermitian manifold with an almost complex structure J. Recently, Sahin [2] proved that every totally umbilical proper slant submanifold of a Kaehler manifold is totally geodesic. Whereas the notion of semi-slant submanifolds of Kaehler manifolds was initiated by Papaghiuc [3]. Bislant submanifolds of an almost Hermitian manifold were introduced as a natural generalization of semi-slant submanifolds by Carriazo [4]. The class of bislant submanifolds includes complex, totally real and CR submanifolds. One of the important classes of bislant submanifolds is that of antislant submanifolds which are studied by Carriazo, but the name antislant seems to refer that it has no slant factor, so Sahin named these submanifolds as hemi-slant submanifolds and studied their warped product in Kaehler setting [5]. In this paper, we study totally umbilical hemi-submanifolds of a Kaehler manifold. In fact, we obtain some classification results for totally umbilical hemi-slant submanifolds of a Kaehler manifold.

2. Preliminaries

Let M̅ be a Riemannian manifold with an almost complex structure J and Hermitian metric g satisfying (a)J2=-I,(b)g(JX,JY)=g(X,Y)
for any X,Y∈TM̅, where TM̅ is the tangent bundle of M̅. If the almost complex structure J satisfies (∇̅XJ)Y=0
for any X,Y∈TxM̅, and x∈M̅, where ∇̅ is the Levi-Civita connection on TM̅, then M̅ is said to have a Kaehler structure and with the structure equation (2.2), an almost Hermitian manifold M̅ is called a Kaehler manifold.

Let M̅ be a Kaehler manifold with almost complex structure J and let M be a Riemannian manifold isometrically immersed in M̅. Then M is called holomorphic (or complex) if J(TxM)⊂TxM, for any x∈M where TxM denotes the tangent space of M at the point x∈M, and totally realy if J(TxM)⊂Tx⊥M, for every x∈M, where Tx⊥M denotes the normal space of M at the point x∈M. There are three other important classes of submanifolds of a Kaehler manifold determined by the behavior of the tangent bundle of the submanifold under the action of the almost complex structure of the ambient manifold.

A submanifold M is called CR submanifold [6] if there exists a differentiable distribution D:x→Dx⊂TxM such that D is invariant with respect to J and its orthogonal complementary distribution D⊥ is antiinvariant with respect to J.

A submanifold M is called slant [1] if for any nonzero vector X tangent to M the angle θ(X) between JX and TxM is constant, that is, it does not depend on the choice of x∈M and X∈TxM.

A submanifold M is called semi-slant [3] if it is endowed with a pair of orthogonal distribution D and Dθ such that D is invariant with respect to J and Dθ is slant, that is, the angle between JX and Dxθ is constant for any X∈Dxθ.

It is clear that the holomorphic and totally real submanifolds are CR submanifolds (respectively, slant submanifolds) with D⊥={0} (resp., θ=0) and D={0} (resp., θ=π/2), respectively. Also, it is clear that CR submanifolds and slant submanifolds are semi slant submanifolds with θ=π/2 and D={0}, respectively.

For an arbitrary submanifold M of a Riemannian manifold M̅ the Gauss and Weingarten formulae are, respectively, given by ∇̅XY=∇XY+h(X,Y),∇̅XN=-ANX+∇X⊥N
for any X,Y∈TM, where ∇ is the induced Riemannian connection on M, N is the vector field normal to M, h is the second fundamental form of M,∇⊥ is the normal connection in the normal bundle T⊥M, and AN is the shape operator of the second fundamental form. Moreover, we have g(ANX,Y)=g(h(X,Y),N),
where g denotes the Riemannian metric on M̅ as well as the metric induced on M. The mean curvature vector H on M is given by H=1n∑i=1nh(ei,ej),
where n is the dimension of M and {e1,e2,…,en} is a local orthonormal frame of vector fields on M.

A submanifold M of a Riemannian manifold M̅ is said to be totally umbilical if h(X,Y)=g(X,Y)H.
If h(X,Y)=0 for any X,Y∈TM then M is said to be totally geodesic submanifold. If H=0, then it is called minimal submanifold.

For any X∈TM we write JX=TX+FX,
where TX and FX are the tangential and normal components of JX, respectively. Similarly, for any vector field N normal to M, we put JN=tN+fN,
where tN and fN are the tangential and normal components of JN, respectively.

The covariant differentiation of J is defined as (∇̅XJ)Y=∇̅XJY-J∇̅XY,
for all X,Y∈TM̅. Similarly, the covariant derivatives of T,F,t and f are (∇̅XT)Y=∇XTY-T∇XY,(∇̅XF)Y=∇X⊥FY-F∇XY,(∇̅Xt)N=∇XtN-t∇X⊥N,(∇̅Xf)N=∇X⊥fN-f∇X⊥N
for any X,Y∈TM, and N∈T⊥M.

On the other hand the covariant derivative of the second fundamental form h is defined as (∇̅Xh)(Y,Z)=∇X⊥h(Y,Z)-h(∇XY,Z)-h(Y,∇XZ)
for any X,Y,Z∈TM. Let R̅ and R be the curvature tensors of the connections ∇̅ and ∇ on M̅ and M, respectively. Then the equations of Gauss and Codazzi are given by R̅(X,Y,Z;W)=R(X,Y,Z;W)-g(h(X,Z),h(Y,W))+g(h(X,W),h(Y,Z)),[R̅(X,Y)Z]⊥=(∇̅Xh)(Y,Z)-(∇̅Yh)(X,Z).

It is known that a submanifold M is slant if and only if T2=λI
for some real number λ∈[-1,0], where I is the Identity transformation of the tangent bundle TM of the submanifold M. Moreover, if M is a slant submanifold and θ is the slant angle of M, then λ=-cos2θ [1].

Hence, for a slant submanifold we have the following relations which are the consequences of (2.15): g(TX,TX)=cos2θg(X,Y),g(FX,FY)=sin2θg(X,Y)
for any X,Y∈TM.

Now, we define the hemi-slant submanifold of an almost Hermitian manifold as follows.

Definition 2.1.

A submanifold M of an almost Hermitian manifold M̅ is said to be a hemi-slant submanifold if there exist two orthogonal complementary distributions Dθ and D⊥ satisfying

TM=Dθ⊕D⊥,

Dθ is a slant distribution with slant angle θ≠π/2,

D⊥ is totally real, that is, JD⊥⊆T⊥M.

It is clear that CR-submanifolds and slant submanifolds are hemi-slant submanifolds with θ=π/2 and Dθ={0}, respectively.

If μ is the invariant subspace under the almost complex structure J of the normal bundle T⊥M, then in the case of pseudoslant submanifold, the normal bundle T⊥M can be decomposed as follows: T⊥M=μ⊕FDθ⊕FD⊥.

On a submanifold M of a Kaehler manifold M̅, by (2.2), (2.3), and (2.7)–(2.11), we have (∇̅XT)Y=AFYX+th(X,Y),(∇̅XF)Y=fh(X,Y)-h(X,TY)
for any X,Y∈TM.

3. Totally Umbilical Hemi-Slant Submanifolds

In this section, we study a special class of hemi-slant submanifolds which are totally umbilical. Throughout the section we consider M as a totally umbilical hemi-slant submanifold of a Kaehler manifold M̅. On a Kaehler manifold M̅, we have the following relations [7]: (a)R̅(JX,JY)Z=R̅(X,Y)Z(b)R̅(X,Y)JZ=JR̅(X,Y)Z
for any X,Y,Z∈TM̅. The Gauss and Weigarten formulae for totally umbilical submanifold of an almost Hermitian manifold are given by∇̅XY=∇XY+g(X,Y)H,∇̅XN=-Xg(H,N)+∇X⊥N
for any X,Y∈TM and N∈T⊥M, where H is the mean curvature vector on M. Also, the Codazzi equation for a totally umbilical submanifold is given by R̅(X,Y,Z,N)=g(Y,Z)g(∇X⊥H,N)-g(X,Z)g(∇Y⊥H,N).

In the following thoerem we consider M as a totally umbilical hemi-slant submanifold with the slant distribution Dθ and totally real distribution D⊥.

Theorem 3.1.

Let M be a totally umbilical hemi-slant submanifold of a Kaehler manifold M̅ such that the mean curvature vector H∈μ. Then one of the following statements is true:

M is totally geodesic in M̅,

M is a CR submanifold of M̅,

M is a totally real submanifold of M̅.

Proof.

For any N∈JD⊥ and X∈Dθ, we have
∇̅XJN=J∇̅XN.
Using (3.2) and (3.3), we obtain
∇XJN+g(X,JN)H=-JXg(H,N)+J∇X⊥N.
Then by orthogonality of two distributions and the fact that H∈μ, the above equation becomes
∇XJN=J∇X⊥N,
which implies that ∇X⊥N∈JD⊥, for any N∈JD⊥. Also, we have g(N,H)=0, for N∈JD⊥, then using this fact, we derive
g(∇X⊥N,H)=-g(N,∇X⊥H)=0.
Then (3.8), gives ∇X⊥H∈μ⊕FDθ. Now, for any X∈Dθ, we have
∇̅XJH=J∇̅XH.
Using (3.3), we obtain
-Xg(H,JH)+∇X⊥JH=-JXg(H,H)+J∇X⊥H.
Since H and JH are orthogonal, then from (2.7), the above equation takes the form
∇X⊥JH=g(H,H)TX+g(H,H)FX+J∇X⊥H.
Taking the product with FX∈FDθ and using (2.17), we obtain
g(∇X⊥JH,FX)=sin2θ‖H‖2‖X‖2+g(F∇⊥H,FX).
Then from (2.17), the last term of right hand side is identically zero using the fact that ∇X⊥H is normal vector and X∈Dθ. Thus, the above equation becomes
g(∇X⊥JH,FX)-sin2θ‖H‖2‖X‖2=0.
Therefore, (3.13) has a solution if either H=0, that is, M is totally geodesic or the angle of slant distribution Dθ is θ=0, that is, M is CR-submanifold or if H≠0, then Dθ={0}, that is, M is a totally real submanifold.

Now, for any X,Y∈TM, by (2.19), we have (∇̅XT)Y=AFYX+th(X,Y).
In particular, if Z∈D⊥, the above equation takes the from -T∇ZZ=AFZZ+th(Z,Z).
Then taking the product with W∈D⊥, we get -g(T∇ZZ,W)=g(AFZZ,W)+g(th(Z,Z),W).
As M is a totally umbilical hemi-slant submanifold, then the above equation takes the form g(Z,W)g(H,FZ)+g(tH,W)‖Z‖2.
Thus, (3.17) has a solution if either H∈μ or dimD⊥=1 or D⊥={0}. If H∉μ then dimD⊥=1 or D⊥={0}.

Remark 3.2.

For a totally umbilical hemi-slant submanifold, if we take H∉μ and D⊥={0}, then the submanifold M is proper slant. Sahin [2] proved that for a totally umbilical proper slant submanifold the mean curvature vector H∈μ. Thus, in case of hemi-slant submanifold we can not take H∉μ and D⊥=0, simultaneously.

Theorem 3.3.

Let M be a totally umbilical hemi-slant submanifold of a Kaehler manifold M̅ such that the dimension of the slant distribution dimDθ≥4 and F is parallel. Then

either M is an extrinsic sphere,

or M is totally real.

Proof.

Since dimDθ≥4, then we can choose a set of orthogonal vectors X,Y∈Dθ, such that g(X,Y)=0. Now from (3.1)(b), we have
JR̅(X,Y)Z=R̅(X,Y)JZ.
Replacing Z by TY, we obtain
JR̅(X,Y)TY=R̅(X,Y)JTY.
Using (2.7) and (2.15), the above equation gives
JR̅(X,Y)TY=-cos2θR̅(X,Y)Y+R̅(X,Y)FTY.
On the other hand, since F is parallel, then we have
R̅(X,Y)FTY=FR̅(X,Y)TY.
Then from (3.20) and (3.21), we obtain
JR̅(X,Y)TY=-cos2θR̅(X,Y)Y+FR̅(X,Y)TY.
Taking the product in (3.22) with N∈T⊥M, we get
g(JR̅(X,Y)TY,N)=-cos2θR̅(X,Y,Y,N)+g(FR̅(X,Y)TY,N).
That is,
cos2θR̅(X,Y,Y,N)=0.
Then from (3.4), we derive
cos2θg(Y,Y)g(∇X⊥H,N)-g(X,Y)g(∇Y⊥H,N)=0.
Since X and Y are orthogonal vectors, then the above equation gives
cos2θg(∇X⊥H,N)‖Y‖2=0.
Therefore, (3.26) gives either θ=π/2 that is, M is totally real or ∇X⊥H=0, for all X∈Dθ. On the same line if we consider X∈D⊥, then we can deduce that either θ=π/2 or ∇X⊥H=0. This means that either M is totally real or ∇X⊥H=0for allX∈TM, that is, the mean curvature vector H is parallel to the submanifold, thus M is an extrinsic sphere.

In our further study, we need the following theorem proved by Yamaguchi et al. [8].

Theorem 3.4.

A complete and simply connected extrinsic sphere Mn in a Kaehler manifold M̅2m is one of the following:

Mn is isometric to an ordinary sphere

Mn is homothetic to a Sasakian manifold

Mn is totally real submanifold and the f-structure is not parallel in the normal bundle.

Now, we are in position to prove our main theorem.

Theorem 3.5.

Let M be a complete simply connected totally umbilical hemi-slant submanifold of a Kaehler manifold M̅. Then M is one of the following:

a totally real submanifold,

a totally geodesic,

a CR submanifold,

dimD⊥=1,

D⊥≠{0},

M is isometric to an ordinary sphere,

M is homothetic to a Sasakian manifold.

The cases (vi) and (vii) hold when F is parallel on M and dimM is odd and ≥5.Proof.

If H∈μ, then by Theorem 3.1, the parts (i), (ii), and (iii) hold. If H∉μ, then (3.17) has a solution if either dimD⊥=1 or D⊥={0} which is case (iv) and we cannot take D⊥={0} and H∉μ, simultaneously for a totally umbilical hemi-slant submanifold due to Remark 3.2 which is case (v). Moreover, H∉μ and F is parallel on M, then by Theorems 3.3 and 3.4, parts (vi) and (vii) hold. This completes the proof of the theorem.

Now, we construct an example of a hemi-slant submanifold in a Kaehler manifold.

Example 3.6.

Consider a submanifold M of ℝ6 with its usual Kaehler structure as
x1=ukcosθ1,x2=uksinθ1,x3=uk,x4=t,x5=t,x6=0,u≠0,k≠0.
The tangent space TM is spanned by the vectors
e1=-uksinθ1∂∂x1+ukcosθ1∂∂x2,e2=1kcosθ1∂∂x1+1ksinθ1∂∂x2+1k∂∂x3,e3=∂∂x4+∂∂x5.
Furthermore, we see that Je3 is orthogonal to TM. If we consider D⊥ and Dθ are the totally real and slant distributions of M, respectively, then D⊥=span{e3} and Dθ=span{e1,e2}. Thus, M is a hemi-slant submanifold of ℝ6 with slant angle θ=cos-1(1/k).

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