Totally Umbilical Hemi-Slant Submanifolds of Kaehler Manifolds

Abstract

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 𝑀 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)(2.1) for any 𝑋,𝑌∈𝑇𝑀, where 𝑇𝑀 is the tangent bundle of 𝑀. If the almost complex structure 𝐽 satisfies ∇_𝑋𝐽𝑌=0(2.2) for any 𝑋,𝑌∈𝑇_𝑥𝑀, and 𝑥∈𝑀, where ∇ is the Levi-Civita connection on 𝑇𝑀, then 𝑀 is said to have a Kaehler structure and with the structure equation (2.2), an almost Hermitian manifold 𝑀 is called a Kaehler manifold.

Let 𝑀 be a Kaehler manifold with almost complex structure 𝐽 and let 𝑀 be a Riemannian manifold isometrically immersed in 𝑀. Then 𝑀 is called holomorphic (or complex) if 𝐽(𝑇_𝑥𝑀)⊂𝑇_𝑥𝑀, for any 𝑥∈𝑀 where 𝑇_𝑥𝑀 denotes the tangent space of 𝑀 at the point 𝑥∈𝑀, and totally realy if 𝐽(𝑇_𝑥𝑀)⊂𝑇^⟂𝑥𝑀, for every 𝑥∈𝑀, where 𝑇^⟂𝑥𝑀 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.(i)A submanifold 𝑀 is called CR submanifold [6] if there exists a differentiable distribution 𝐷∶𝑥→𝐷_𝑥⊂𝑇_𝑥𝑀 such that 𝐷 is invariant with respect to 𝐽 and its orthogonal complementary distribution 𝐷^⟂ is antiinvariant with respect to 𝐽.(ii)A submanifold 𝑀 is called slant [1] if for any nonzero vector 𝑋 tangent to 𝑀 the angle 𝜃(𝑋) between 𝐽𝑋 and 𝑇_𝑥𝑀 is constant, that is, it does not depend on the choice of 𝑥∈𝑀 and 𝑋∈𝑇_𝑥𝑀.(iii)A submanifold 𝑀 is called semi-slant [3] if it is endowed with a pair of orthogonal distribution 𝐷 and 𝐷^𝜃 such that 𝐷 is invariant with respect to 𝐽 and 𝐷^𝜃 is slant, that is, the angle between 𝐽𝑋 and 𝐷^𝜃𝑥 is constant for any 𝑋∈𝐷^𝜃𝑥.

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 𝑀 the Gauss and Weingarten formulae are, respectively, given by ∇_𝑋𝑌=∇_𝑋𝑌+ℎ(𝑋,𝑌),∇_𝑋𝑁=−𝐴_𝑁𝑋+∇^⟂𝑋𝑁(2.3) 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 𝐴_𝑁 is the shape operator of the second fundamental form. Moreover, we have 𝑔𝐴_𝑁𝑋,𝑌=𝑔(ℎ(𝑋,𝑌),𝑁),(2.4) where 𝑔 denotes the Riemannian metric on 𝑀 as well as the metric induced on 𝑀. The mean curvature vector 𝐻 on 𝑀 is given by 𝐻=1𝑛^𝑛_𝑖=1ℎ𝑒_𝑖,𝑒_𝑗,(2.5) where 𝑛 is the dimension of 𝑀 and {𝑒₁,𝑒₂,…,𝑒_𝑛} is a local orthonormal frame of vector fields on 𝑀.

A submanifold 𝑀 of a Riemannian manifold 𝑀 is said to be totally umbilical if ℎ(𝑋,𝑌)=𝑔(𝑋,𝑌)𝐻.(2.6) 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,(2.7) 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,(2.8) 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,(2.9) for all 𝑋,𝑌∈𝑇𝑀. Similarly, the covariant derivatives of 𝑇,𝐹,𝑡 and 𝑓 are ∇_𝑋𝑇𝑌=∇_𝑋𝑇𝑌−𝑇∇_𝑋𝑌,(2.10)∇_𝑋𝐹𝑌=∇^⟂𝑋𝐹𝑌−𝐹∇_𝑋𝑌,(2.11)∇_𝑋𝑡𝑁=∇_𝑋𝑡𝑁−𝑡∇^⟂𝑋𝑁,∇_𝑋𝑓𝑁=∇^⟂𝑋𝑓𝑁−𝑓∇^⟂𝑋𝑁(2.12) 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)=∇^⟂𝑋ℎ(𝑌,𝑍)−ℎ∇_𝑋𝑌,𝑍−ℎ𝑌,∇_𝑋𝑍(2.13) for any 𝑋,𝑌,𝑍∈𝑇𝑀. Let 𝑅 and 𝑅 be the curvature tensors of the connections ∇ and ∇ on 𝑀 and 𝑀, respectively. Then the equations of Gauss and Codazzi are given by 𝑅(𝑋,𝑌,𝑍;𝑊)=𝑅(𝑋,𝑌,𝑍;𝑊)−𝑔(ℎ(𝑋,𝑍),ℎ(𝑌,𝑊))+𝑔(ℎ(𝑋,𝑊),ℎ(𝑌,𝑍)),𝑅(𝑋,𝑌)𝑍^⟂=∇_𝑋ℎ(𝑌,𝑍)−∇_𝑌ℎ(𝑋,𝑍).(2.14)

It is known that a submanifold 𝑀 is slant if and only if 𝑇²=𝜆𝐼(2.15) 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): 𝑔𝑔((𝑇𝐹𝑋𝑋,,𝑇𝐹𝑋𝑌))==csoisn²₂𝜃𝑔(𝑋,𝑌),(2.16)𝜃𝑔(𝑋,𝑌)(2.17) 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 𝑀 is said to be a hemi-slant submanifold if there exist two orthogonal complementary distributions 𝐷^𝜃 and 𝐷^⟂ satisfying (i)𝑇𝑀=𝐷^𝜃⊕𝐷^⟂, (ii)𝐷^𝜃 is a slant distribution with slant angle 𝜃≠𝜋/2,(iii)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 𝐷^𝜃={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⊕𝐹𝐷^𝜃⊕𝐹𝐷^⟂.(2.18)

On a submanifold 𝑀 of a Kaehler manifold 𝑀, by (2.2), (2.3), and (2.7)–(2.11), we have ∇_𝑋𝑇𝑌=𝐴_𝐹𝑌𝑋+𝑡ℎ(𝑋,𝑌),(2.19)∇_𝑋𝐹𝑌=𝑓ℎ(𝑋,𝑌)−ℎ(𝑋,𝑇𝑌)(2.20) 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 𝑀. On a Kaehler manifold 𝑀, we have the following relations [7]: (a)𝑅(𝐽𝑋,𝐽𝑌)𝑍=𝑅(𝑋,𝑌)𝑍(b)𝑅(𝑋,𝑌)𝐽𝑍=𝐽𝑅(𝑋,𝑌)𝑍(3.1) for any 𝑋,𝑌,𝑍∈𝑇𝑀. The Gauss and Weigarten formulae for totally umbilical submanifold of an almost Hermitian manifold are given by∇_𝑋𝑌=∇_𝑋𝑌+𝑔(𝑋,𝑌)𝐻,(3.2)∇_𝑋𝑁=−𝑋𝑔(𝐻,𝑁)+∇^⟂𝑋𝑁(3.3) 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 𝑅(𝑋,𝑌,𝑍,𝑁)=𝑔(𝑌,𝑍)𝑔∇^⟂𝑋𝐻,𝑁−𝑔(𝑋,𝑍)𝑔∇^⟂𝑌𝐻,𝑁.(3.4)

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

Theorem 3.1. Let 𝑀 be a totally umbilical hemi-slant submanifold of a Kaehler manifold 𝑀 such that the mean curvature vector 𝐻∈𝜇. Then one of the following statements is true: (i)𝑀 is totally geodesic in 𝑀,(ii)𝑀 is a CR submanifold of 𝑀,(iii)𝑀 is a totally real submanifold of 𝑀.

Proof. For any 𝑁∈𝐽𝐷^⟂ and 𝑋∈𝐷^𝜃, we have ∇_𝑋𝐽𝑁=𝐽∇_𝑋𝑁.(3.5) Using (3.2) and (3.3), we obtain ∇_𝑋𝐽𝑁+𝑔(𝑋,𝐽𝑁)𝐻=−𝐽𝑋𝑔(𝐻,𝑁)+𝐽∇^⟂𝑋𝑁.(3.6) Then by orthogonality of two distributions and the fact that 𝐻∈𝜇, the above equation becomes ∇_𝑋𝐽𝑁=𝐽∇^⟂𝑋𝑁,(3.7) which implies that ∇^⟂𝑋𝑁∈𝐽𝐷^⟂, for any 𝑁∈𝐽𝐷^⟂. Also, we have 𝑔(𝑁,𝐻)=0, for 𝑁∈𝐽𝐷^⟂, then using this fact, we derive 𝑔∇^⟂𝑋𝑁,𝐻=−𝑔𝑁,∇^⟂𝑋𝐻=0.(3.8) Then (3.8), gives ∇^⟂𝑋𝐻∈𝜇⊕𝐹𝐷^𝜃. Now, for any 𝑋∈𝐷^𝜃, we have ∇_𝑋𝐽𝐻=𝐽∇_𝑋𝐻.(3.9) Using (3.3), we obtain −𝑋𝑔(𝐻,𝐽𝐻)+∇^⟂𝑋𝐽𝐻=−𝐽𝑋𝑔(𝐻,𝐻)+𝐽∇^⟂𝑋𝐻.(3.10) Since 𝐻 and 𝐽𝐻 are orthogonal, then from (2.7), the above equation takes the form ∇^⟂𝑋𝐽𝐻=𝑔(𝐻,𝐻)𝑇𝑋+𝑔(𝐻,𝐻)𝐹𝑋+𝐽∇^⟂𝑋𝐻.(3.11) Taking the product with 𝐹𝑋∈𝐹𝐷^𝜃 and using (2.17), we obtain 𝑔∇^⟂𝑋𝐽𝐻,𝐹𝑋=sin²𝜃‖𝐻‖²‖𝑋‖²+𝑔𝐹∇^⟂𝐻,𝐹𝑋.(3.12) Then from (2.17), the last term of right hand side is identically zero using the fact that ∇^⟂𝑋𝐻 is normal vector and 𝑋∈𝐷^𝜃. Thus, the above equation becomes 𝑔∇^⟂𝑋𝐽𝐻,𝐹𝑋−sin²𝜃‖𝐻‖²‖𝑋‖²=0.(3.13) Therefore, (3.13) has a solution if either 𝐻=0, that is, 𝑀 is totally geodesic or the angle of slant distribution 𝐷^𝜃 is 𝜃=0, that is, 𝑀 is CR-submanifold or if 𝐻≠0, then 𝐷^𝜃={0}, that is, 𝑀 is a totally real submanifold.

Now, for any 𝑋,𝑌∈𝑇𝑀, by (2.19), we have ∇_𝑋𝑇𝑌=𝐴_𝐹𝑌𝑋+𝑡ℎ(𝑋,𝑌).(3.14) In particular, if 𝑍∈𝐷^⟂, the above equation takes the from −𝑇∇_𝑍𝑍=𝐴_𝐹𝑍𝑍+𝑡ℎ(𝑍,𝑍).(3.15) Then taking the product with 𝑊∈𝐷^⟂, we get −𝑔𝑇∇_𝑍𝑍,𝑊=𝑔𝐴_𝐹𝑍𝑍,𝑊+𝑔(𝑡ℎ(𝑍,𝑍),𝑊).(3.16) 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.(3.17) 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 𝑀 such that the dimension of the slant distribution dim𝐷^𝜃≥4 and 𝐹 is parallel. Then (i)either 𝑀 is an extrinsic sphere,(ii)or 𝑀 is totally real.

Proof. Since dim𝐷^𝜃≥4, then we can choose a set of orthogonal vectors 𝑋,𝑌∈𝐷^𝜃, such that 𝑔(𝑋,𝑌)=0. Now from (3.1)(b), we have 𝐽𝑅(𝑋,𝑌)𝑍=𝑅(𝑋,𝑌)𝐽𝑍.(3.18) Replacing 𝑍 by 𝑇𝑌, we obtain 𝐽𝑅(𝑋,𝑌)𝑇𝑌=𝑅(𝑋,𝑌)𝐽𝑇𝑌.(3.19) Using (2.7) and (2.15), the above equation gives 𝐽𝑅(𝑋,𝑌)𝑇𝑌=−cos²𝜃𝑅(𝑋,𝑌)𝑌+𝑅(𝑋,𝑌)𝐹𝑇𝑌.(3.20) On the other hand, since 𝐹 is parallel, then we have 𝑅(𝑋,𝑌)𝐹𝑇𝑌=𝐹𝑅(𝑋,𝑌)𝑇𝑌.(3.21) Then from (3.20) and (3.21), we obtain 𝐽𝑅(𝑋,𝑌)𝑇𝑌=−cos²𝜃𝑅(𝑋,𝑌)𝑌+𝐹𝑅(𝑋,𝑌)𝑇𝑌.(3.22) Taking the product in (3.22) with 𝑁∈𝑇^⟂𝑀, we get 𝑔𝐽𝑅(𝑋,𝑌)𝑇𝑌,𝑁=−cos²𝜃𝑅(𝑋,𝑌,𝑌,𝑁)+𝑔𝐹𝑅(𝑋,𝑌)𝑇𝑌,𝑁.(3.23) That is, cos²𝜃𝑅(𝑋,𝑌,𝑌,𝑁)=0.(3.24) Then from (3.4), we derive cos²𝜃𝑔(𝑌,𝑌)𝑔∇^⟂𝑋𝐻,𝑁−𝑔(𝑋,𝑌)𝑔∇^⟂𝑌𝐻,𝑁=0.(3.25) Since 𝑋 and 𝑌 are orthogonal vectors, then the above equation gives cos²𝜃𝑔∇^⟂𝑋𝐻,𝑁‖𝑌‖²=0.(3.26) Therefore, (3.26) gives either 𝜃=𝜋/2 that is, 𝑀 is totally real or ∇^⟂𝑋𝐻=0, for all 𝑋∈𝐷^𝜃. On the same line if we consider 𝑋∈𝐷^⟂, then we can deduce that either 𝜃=𝜋/2 or ∇^⟂𝑋𝐻=0. This means that either 𝑀 is totally real or ∇^⟂𝑋𝐻=0forall𝑋∈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 𝑀^𝑛 in a Kaehler manifold 𝑀^2𝑚 is one of the following: (i)𝑀^𝑛 is isometric to an ordinary sphere(ii)𝑀^𝑛 is homothetic to a Sasakian manifold(iii)𝑀^𝑛 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 𝑀. Then 𝑀 is one of the following: (i)a totally real submanifold,(ii)a totally geodesic,(iii)a CR submanifold,(iv)dim𝐷^⟂=1,(v)𝐷^⟂≠{0},(vi)𝑀 is isometric to an ordinary sphere,(vii)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=𝑢𝑥₄𝑘,=𝑡,𝑥₅=𝑡,𝑥₆=0,𝑢≠0,𝑘≠0.(3.27) The tangent space 𝑇𝑀 is spanned by the vectors 𝑒₁=−𝑢𝑘sin𝜃¹𝜕𝜕𝑥₁+𝑢𝑘cos𝜃¹𝜕𝜕𝑥₂,𝑒₂=1𝑘cos𝜃¹𝜕𝜕𝑥₁+1𝑘sin𝜃¹𝜕𝜕𝑥₂+1𝑘𝜕𝜕𝑥₃𝑒₃,=𝜕𝜕𝑥₄+𝜕𝜕𝑥₅.(3.28) Furthermore, we see that 𝐽𝑒₃ is orthogonal to 𝑇𝑀. If we consider 𝐷^⟂ and 𝐷^𝜃 are the totally real and slant distributions of 𝑀, respectively, then 𝐷^⟂=span{𝑒₃} and 𝐷^𝜃=span{𝑒₁,𝑒₂}. Thus, 𝑀 is a hemi-slant submanifold of ℝ⁶ with slant angle 𝜃=cos⁻1(1/𝑘).