Some classes of CR submanifolds with an umbilical section of the nearly Kähler
Introduction
An almost Hermitian manifold , with Levi-Civita connection , is called a nearly Kähler manifold if for any tangent vector it holds . It is known that there exist only four six-dimensional homogeneous nearly Kähler manifolds, that are not Kähler: the sphere , the complex projective space , the flag manifold and , see [5]. A submanifold of manifold is called a submanifold if there exists a invariant distribution on , such that for its orthogonal complement in it holds . Further results about submanifolds may be found in [3].
Recall that for tangent vector fields and normal , the formulas of Gauss and Weingarten are given by and , where are, respectively, the induced connection of the submanifold and the connection in the normal bundle, while and are the second fundamental form of submanifold and the shape operator. If for a normal section on , is everywhere proportional to the identity transformation , that is, , for some function , then is called an umbilical section on , or is said to be umbilical with respect to . If the submanifold is umbilical with respect to every local normal section in , then is said to be totally umbilical. For a -dimensional submanifold in -dimensional manifold , the mean curvature vector is defined by , where vectors form an orthonormal frame of . If vanishes identically then the submanifold is called a minimal submanifold. A distribution on is totally geodesic if and only if the second fundamental form restricted to vector fields belonging to vanishes identically.
Section snippets
Preliminaries
A unit sphere in the space can be identified with the space of quaternions and by using the isomorphism of the spaces an arbitrary tangent vector at a point is represented by , where and are imaginary quaternions. The almost complex structure is an endomorphism such that and on is defined as and the Hermitian metric is given by for .
Three-dimensional CR submanifolds of the nearly Kähler
For an orthonormal moving frame along a three-dimensional proper CR submanifold we can take vector fields and that span the two-dimensional distribution and that spans . Then for an orthonormal frame of we can take vector fields , and . For these vector fields the following equalities hold: , , , , , , , , ,
Four-dimensional CR submanifolds of the nearly Kähler
Let be a four-dimensional proper CR submanifold implying that the distributions and are both two-dimensional. First we will construct a local moving frame suitable for computation. We can take unit vector fields and that span , and mutually orthogonal vector fields and that span . Then the normal distribution is spanned by the vector fields and . By using (1) the following relations are obtained , ,
Existence of four-dimensional CR submanifolds of that have an umbilical section
Theorem 9 There is no four-dimensional CR submanifold of the nearly Kähler that has an umbilical section.
Proof By applying a rotation in the distribution , we can assume that the vector field is an umbilical section. Now, we will investigate a four-dimensional CR submanifold with vector field as umbilical section. By a definition of an umbilical section it holds , , , , , , , , . As the distributions and are mutually
Umbilical three-dimensional CR submanifolds of
By a definition of an umbilical submanifold, if a dimension of the CR submanifold of is three, by a relations (2), the following equations hold: for some umbilical section . Directly, from this equations and relations given by Lemma 2 we get that there are no totally umbilical three-dimensional CR submanifolds. Directly we obtain that .
Umbilical three-dimensional CR submanifolds of with the totally geodesic distribution
Here, we assume that the distribution is totally geodesic implying , , so we have that . For the almost product structure expressed as (4), from Eq. (3) evaluated for the vector fields , we obtain , as the coefficients , cannot vanish at same time.
Lemma 10 There is no three-dimensional CR submanifold of the , with totally geodesic the almost complex distribution , that admits an umbilical section belonging to the
Classification of the three-dimensional umbilical CR submanifolds of the such that
This section contains some results useful for further investigation. It contains an example of an umbilical submanifold which an almost complex distribution is integrable and at the same time it is an example of an umbilical submanifold with vanishing function . Also, a condition implies that .
Theorem 13 The three-dimensional umbilical CR submanifold of the such that is locally isometric to a submanifold given by Theorem (5) for which it holds and umbilical vector field
Some umbilical three-dimensional CR submanifolds of that are minimal
In this section submanifold is minimal and by definition of a minimal submanifold it holds , , .
Theorem 14 Let be a minimal three-dimensional CR submanifold of that is umbilical with respect to a vector field . Then is locally isometric to the immersion given by Theorem (4) or one of the following three immersions:
Acknowledgement
Funding: The research was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia [no. 451-03-68/2020-14/200116].
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