Some classes of CR submanifolds with an umbilical section of the nearly Kähler S3×S3

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Abstract

Recently, the investigation of a CR submanifolds of the nearly Kähler manifold S3×S3 was started. In this paper it is proved that CR submanifolds of the nearly Kähler manifold S3×S3 with umbilical sections must have dimension three and then we obtain some examples of them with distinguished vector fields. Also, we classify minimal submanifolds that have a vector field E4 as an umbilical section. The main result is classification of the three-dimensional umbilical CR submanifolds with totally geodesic and almost complex distribution D1.

Introduction

An almost Hermitian manifold (M˜,g,J), with Levi-Civita connection ˜, is called a nearly Kähler manifold if for any tangent vector X it holds (˜XJ)X=0. It is known that there exist only four six-dimensional homogeneous nearly Kähler manifolds, that are not Kähler: the sphere S6, the complex projective space P3, the flag manifold F3 and S3×S3, see [5]. A submanifold M of manifold (M˜,g,J) is called a CR submanifold if there exists a J invariant distribution D1 on M, such that for its orthogonal complement D1 in TM it holds J(D1)=TM. Further results about CR submanifolds may be found in [3].

Recall that for tangent vector fields X,Y and normal ξ, the formulas of Gauss and Weingarten are given by ˜XY=XY+h(X,Y) and ˜Xξ=AξX+Xξ, where , are, respectively, the induced connection of the submanifold and the connection in the normal bundle, while h and A are the second fundamental form of submanifold M and the shape operator. If for a normal section ξ on M, Aξ is everywhere proportional to the identity transformation I, that is, Aξ=ρI, for some function ρ, then ξ is called an umbilical section on M, or M is said to be umbilical with respect to ξ. If the submanifold M is umbilical with respect to every local normal section in M, then M is said to be totally umbilical. For a n-dimensional submanifold M in m-dimensional manifold M˜, the mean curvature vector is defined by H=1ni=1nh(Ei,Ei), where vectors Ei form an orthonormal frame of TM. If H vanishes identically then the submanifold M is called a minimal submanifold. A distribution D on M is totally geodesic if and only if the second fundamental form restricted to vector fields belonging to D vanishes identically.

Section snippets

Preliminaries

A unit sphere S3 in the space R4 can be identified with the space of quaternions H and by using the isomorphism of the spaces T(p,q)(S3×S3)TpS3TqS3 an arbitrary tangent vector at a point (p,q)S3×S3 is represented by Z(p,q)=(pα,qβ), where α and β are imaginary quaternions. The almost complex structure J is an endomorphism such that J2=Id and on S3×S3 is defined as J(Z(p,q))=(13)(p(2βα),q(2α+β)) and the Hermitian metric g is given by g(Z,Z)=12(Z,Z+JZ,JZ) for Z,ZT(p,q)(S3×S3).

Three-dimensional CR submanifolds of the nearly Kähler S3×S3

For an orthonormal moving frame along a three-dimensional proper CR submanifold M we can take vector fields E1 and E2=JE1 that span the two-dimensional distribution D1 and E3 that spans D2=D1. Then for an orthonormal frame of TM we can take vector fields E4=JE3, E5=3G(E1,E3) and E6=3G(E2,E3)=JE5. For these vector fields the following equalities hold: G(E1,E2)=0, G(E1,E3)=13E5, G(E1,E4)=13E6, G(E1,E5)=13E3, G(E1,E6)=13E4, G(E2,E3)=13E6, G(E2,E4)=13E5, G(E2,E5)=13E4, G(E2,E6)=13E3,

Four-dimensional CR submanifolds of the nearly Kähler S3×S3

Let M be a four-dimensional proper CR submanifold implying that the distributions D1 and D1 are both two-dimensional. First we will construct a local moving frame suitable for computation. We can take unit vector fields E1 and E2=JE1 that span D1, and mutually orthogonal vector fields E3 and E4 that span D2=D1. Then the normal distribution D3 is spanned by the vector fields E5=JE3 and E6=JE4. By using (1) the following relations are obtained G(E1,E2)=0, G(E1,E3)=cost3E4+sint3E6, G(E1,E4)=

Existence of four-dimensional CR submanifolds of S3×S3 that have an umbilical section

Theorem 9

There is no four-dimensional CR submanifold of the nearly Kähler S3×S3 that has an umbilical section.

Proof

By applying a rotation in the distribution D2, we can assume that the vector field E5 is an umbilical section. Now, we will investigate a four-dimensional CR submanifold with vector field E5 as umbilical section. By a definition of an umbilical section it holds h121=0, h131=0, h132=13cost, h221=h111, h231=0, h232=13sint, h331=h111, h332=0, h342=h111. As the distributions D1 and D2 are mutually

Umbilical three-dimensional CR submanifolds of S3×S3

By a definition of an umbilical submanifold, if a dimension of the CR submanifold of S3×S3 is three, by a relations (2), the following equations hold: ah111+bh112+ch113=ρah121+bh113ch112=0ah131+bh132+ch133=0ah221bh112ch113=ρah231b(13h133)ch132=0ah331+bh332+ch333=ρ for some umbilical section F=aE4+bE5+cE6. 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 ρ=(h111+h221)a2.

Umbilical three-dimensional CR submanifolds of S3×S3 with the totally geodesic distribution D1

Here, we assume that the distribution D1 is totally geodesic implying h(Ei,Ej)=0, {i,j}{1,2}, so we have that h111=h112=h113=h121=h221=0. For the almost product structure P expressed as (4), from Eq. (3) evaluated for the vector fields E1,E2,E1, we obtain cosθsinθ=0, as the coefficients wi, i{1,2,3,4} cannot vanish at same time.

Lemma 10

There is no three-dimensional CR submanifold of the S3×S3, with totally geodesic the almost complex distribution D1, that admits an umbilical section belonging to the

Classification of the three-dimensional umbilical CR submanifolds of the S3×S3 such that PD1D1

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 ρ. Also, a condition PD1D1 implies that PD1=D2.

Theorem 13

The three-dimensional umbilical CR submanifold of the S3×S3 such that PD1D1 is locally isometric to a submanifold given by Theorem (5) for which it holds PD1=D2 and umbilical vector field

Some umbilical three-dimensional CR submanifolds of S3×S3 that are minimal

In this section submanifold M is minimal and by definition of a minimal submanifold it holds h331=h111h221, h332=0, h333=0.

Theorem 14

Let M be a minimal three-dimensional CR submanifold of S3×S3 that is umbilical with respect to a vector field E4. Then M is locally isometric to the immersion given by Theorem (4) or one of the following three immersions: f1(x,y,t)=(((kcos(32z)jsin(32z))cos((3x+y)2)+(cos(32z)+isin(32z))sin((3x+y)2)),((kcos(32z)+jsin(32z))cos((y3x)2)+(cos(32z)+isin(32z))sin((

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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