Research Article | Open Access
Totally Umbilical Hemi-Slant Submanifolds of Kaehler Manifolds
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.
The notion of slant submanifolds of an almost Hermitian manifold was introduced by Chen . These submanifolds are the generalization of both holomorphic and totally real submanifolds of an almost Hermitian manifold with an almost complex structure . Recently, Sahin  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 . Bislant submanifolds of an almost Hermitian manifold were introduced as a natural generalization of semi-slant submanifolds by Carriazo . 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 . 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.
Let be a Riemannian manifold with an almost complex structure and Hermitian metric satisfying for any , where is the tangent bundle of . If the almost complex structure satisfies for any , and , where is the Levi-Civita connection on , then is said to have a Kaehler structure and with the structure equation (2.2), an almost Hermitian manifold is called a Kaehler manifold.
Let be a Kaehler manifold with almost complex structure and let be a Riemannian manifold isometrically immersed in . Then is called holomorphic (or complex) if , for any where denotes the tangent space of at the point , and totally realy if , for every , where denotes the normal space of at the point . 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 is called CR submanifold  if there exists a differentiable distribution such that is invariant with respect to and its orthogonal complementary distribution is antiinvariant with respect to .(ii)A submanifold is called slant  if for any nonzero vector tangent to the angle between and is constant, that is, it does not depend on the choice of and .(iii)A submanifold is called semi-slant  if it is endowed with a pair of orthogonal distribution and such that is invariant with respect to and is slant, that is, the angle between and is constant for any .
It is clear that the holomorphic and totally real submanifolds are CR submanifolds (respectively, slant submanifolds) with (resp., ) and (resp., ), respectively. Also, it is clear that CR submanifolds and slant submanifolds are semi slant submanifolds with and , respectively.
For an arbitrary submanifold of a Riemannian manifold the Gauss and Weingarten formulae are, respectively, given by for any , where is the induced Riemannian connection on , is the vector field normal to , is the second fundamental form of is the normal connection in the normal bundle , and is the shape operator of the second fundamental form. Moreover, we have where denotes the Riemannian metric on as well as the metric induced on . The mean curvature vector on is given by where is the dimension of and is a local orthonormal frame of vector fields on .
A submanifold of a Riemannian manifold is said to be totally umbilical if If for any then is said to be totally geodesic submanifold. If , then it is called minimal submanifold.
For any we write where and are the tangential and normal components of , respectively. Similarly, for any vector field normal to , we put where and are the tangential and normal components of , respectively.
The covariant differentiation of is defined as for all . Similarly, the covariant derivatives of and are for any , and .
On the other hand the covariant derivative of the second fundamental form is defined as for any . Let and be the curvature tensors of the connections and on and , respectively. Then the equations of Gauss and Codazzi are given by
It is known that a submanifold is slant if and only if for some real number , where is the Identity transformation of the tangent bundle of the submanifold . Moreover, if is a slant submanifold and is the slant angle of , then .
Hence, for a slant submanifold we have the following relations which are the consequences of (2.15): for any .
Now, we define the hemi-slant submanifold of an almost Hermitian manifold as follows.
Definition 2.1. A submanifold of an almost Hermitian manifold is said to be a hemi-slant submanifold if there exist two orthogonal complementary distributions and satisfying (i), (ii) is a slant distribution with slant angle ,(iii) is totally real, that is, .It is clear that CR-submanifolds and slant submanifolds are hemi-slant submanifolds with and , respectively.
If is the invariant subspace under the almost complex structure of the normal bundle , then in the case of pseudoslant submanifold, the normal bundle can be decomposed as follows:
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 as a totally umbilical hemi-slant submanifold of a Kaehler manifold . On a Kaehler manifold , we have the following relations : for any . The Gauss and Weigarten formulae for totally umbilical submanifold of an almost Hermitian manifold are given by for any and , where is the mean curvature vector on . Also, the Codazzi equation for a totally umbilical submanifold is given by
In the following thoerem we consider as a totally umbilical hemi-slant submanifold with the slant distribution and totally real distribution .
Theorem 3.1. Let be a totally umbilical hemi-slant submanifold of a Kaehler manifold such that the mean curvature vector . Then one of the following statements is true: (i) is totally geodesic in ,(ii) is a CR submanifold of ,(iii) is a totally real submanifold of .
Proof. For any and , we have Using (3.2) and (3.3), we obtain Then by orthogonality of two distributions and the fact that , the above equation becomes which implies that , for any . Also, we have , for , then using this fact, we derive Then (3.8), gives . Now, for any , we have Using (3.3), we obtain Since and are orthogonal, then from (2.7), the above equation takes the form Taking the product with and using (2.17), we obtain Then from (2.17), the last term of right hand side is identically zero using the fact that is normal vector and . Thus, the above equation becomes Therefore, (3.13) has a solution if either , that is, is totally geodesic or the angle of slant distribution is , that is, is CR-submanifold or if , then , that is, is a totally real submanifold.
Now, for any , by (2.19), we have In particular, if , the above equation takes the from Then taking the product with , we get As is a totally umbilical hemi-slant submanifold, then the above equation takes the form Thus, (3.17) has a solution if either or or . If then or .
Remark 3.2. For a totally umbilical hemi-slant submanifold, if we take and , then the submanifold is proper slant. Sahin  proved that for a totally umbilical proper slant submanifold the mean curvature vector . Thus, in case of hemi-slant submanifold we can not take and , simultaneously.
Theorem 3.3. Let be a totally umbilical hemi-slant submanifold of a Kaehler manifold such that the dimension of the slant distribution and is parallel. Then (i)either is an extrinsic sphere,(ii)or is totally real.
Proof. Since , then we can choose a set of orthogonal vectors , such that . Now from (3.1)(b), we have Replacing by , we obtain Using (2.7) and (2.15), the above equation gives On the other hand, since is parallel, then we have Then from (3.20) and (3.21), we obtain Taking the product in (3.22) with , we get That is, Then from (3.4), we derive Since and are orthogonal vectors, then the above equation gives Therefore, (3.26) gives either that is, is totally real or , for all . On the same line if we consider , then we can deduce that either or . This means that either is totally real or , that is, the mean curvature vector is parallel to the submanifold, thus is an extrinsic sphere.
In our further study, we need the following theorem proved by Yamaguchi et al. .
Theorem 3.4. A complete and simply connected extrinsic sphere in a Kaehler manifold is one of the following: (i) is isometric to an ordinary sphere(ii) is homothetic to a Sasakian manifold(iii) 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 be a complete simply connected totally umbilical hemi-slant submanifold of a Kaehler manifold . Then is one of the following: (i)a totally real submanifold,(ii)a totally geodesic,(iii)a CR submanifold,(iv),(v),(vi) is isometric to an ordinary sphere,(vii) is homothetic to a Sasakian manifold.The cases (vi) and (vii) hold when is parallel on and is odd and .
Proof. If , then by Theorem 3.1, the parts (i), (ii), and (iii) hold. If , then (3.17) has a solution if either or which is case (iv) and we cannot take and , simultaneously for a totally umbilical hemi-slant submanifold due to Remark 3.2 which is case (v). Moreover, and is parallel on , 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 of with its usual Kaehler structure as The tangent space is spanned by the vectors Furthermore, we see that is orthogonal to . If we consider and are the totally real and slant distributions of , respectively, then and . Thus, is a hemi-slant submanifold of with slant angle .
- B. Y. Chen, “Slant immersions,” Bulletin of the Australian Mathematical Society, vol. 41, no. 1, pp. 135–147, 1990.
- B. Sahin, “Every totally umbilical proper slant submanifold of a Kähler manifold is totally geodesic,” Results in Mathematics, vol. 54, no. 1-2, pp. 167–172, 2009.
- N. Papaghiuc, “Semi-slant submanifolds of a Kaehlerian manifold,” Analele Ştiinţific Ale Universitǎţii Iaşi, vol. 9, no. 1, pp. 55–61, 1994.
- A. Carriazo, New Developments in Slant Submanifolds Theory, Narosa Publishing, New Delhi, India, 2002.
- B. Sahin, “Warped product submanifolds of Kaehler manifolds with a slant factor,” Annales Polonici Mathematici, vol. 95, no. 3, pp. 207–226, 2009.
- A. Bejancu, Geometry of CR-Submanifolds, vol. 23 of Mathematics and its Applications, Kluwer Academic publisher, Dodrecht, The Netherlands, 1986.
- M. Obata, “Riemannian manifolds admitting a solution of a certain system of differential equations,” pp. 101–114, Nippon Hyoronsha, 1965.
- S. Yamaguchi, H. Nemoto, and N. Kawabata, “Extrinsic spheres in a Kähler manifold,” Michigan Mathematical Journal, vol. 31, no. 1, pp. 15–19, 1984.
Copyright © 2011 Falleh R. Al-Solamy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.