PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 101(115) (2017), 25–35 https://doi.org/10.2298/PIM1715025A

RULED THREE-DIMENSIONAL CR SUBMANIFOLDS OF THE SPHERE S^{6}(1)

Miroslava Antić

Abstract. We investigate proper, three-dimensional CR submanifolds of the nearly Kähler sphere S^{6}(1) ruled by totally geodesic spheres S^{2}(1), and classify them by using a sphere curve and a vector field along that curve.

1. Introduction

It is well known that by using multiplication of the octonions O and identifying the space of ImO with ℝ^{7}, it is possible to introduce a vector cross product × in the space ℝ^{7}. This cross product induces an almost complex structure J on the standard unit sphere S^{6}(1) in ℝ^{7} which is Hermitian and almost complex, and moreover gives a nearly Kähler structure to S^{6}(1).

Recall that a submanifold M of a manifold with an almost complex structure J is also called almost complex, if its tangent bundle is invariant for J, i.e., JT_{p}M ⊂ T_{p}M, p ∈ M. If JT_{p}M ⊂ T_{p}^{⊥} M, p ∈ M, where T^{⊥} M is a normal bundle of the submanifold, M is called a totally real submanifold. One of the generalizations of almost complex and totally real submanifolds are CR submanifolds. By the definition of Bejancu [2], a submanifold M is called a CR submanifold if there exists on M a differentiable almost complex distribution U such that its orthogonal complement U^{⊥} ⊂ T M is a totally real distribution. If a CR submanifold is neither almost complex, nor totally real, it is a proper CR submanifold. Due to the dimension restrictions it is clear that a proper CR submanifold of the sphere S^{6}(1) can be of dimensions three, four and five. All hypersurfaces of the sphere are trivially CR so the focus of the investigation is mostly on those of dimension three and four. Here, we deal with the three dimensional case. Such CR submanifolds have been previously studied amongst others by K. Mashimo, H. Hashimoto, K. Sekigawa, M. Djorić and L. Vrancken. Particularly in [7] and [6] one of the first known families of the three dimensional minimal CR submanifolds was introduced, and in [3] was obtained the classification of the minimal CR submanifolds which satisfy Chen’s basic equality. We also recall that the submanifold M of a Riemannian manifold (M̃, g) is said to be ruled, if it admits a foliation with leaves that are totally geodesically immersed into (M̃, g). Then, trivially, the second fundamental form vanishes on the distribution corresponding to the foliation. Such distribution is said to be totally geodesic.

2010 Mathematics Subject Classification: Primary 53B25; Secondary 53C42, 53C40.

Key words and phrases: CR submanifold, nearly Kähler six sphere, ruled submanifolds.

Communicated by Stevan Pilipović.

In [1] it was shown that for a three dimensional CR submanifold of S^{6}(1) it is equivalent

(1) the CR submanifold is minimal and contained in a totally geodesic hypersphere

(2) the CR submanifold is U and U^{⊥} totally geodesic and that examples of [6] and [3] are of this type. Moreover, there it was shown that such a submanifold is locally congruent to the immersion

F (s, y_{1}, y_{2}, y_{3}) = y_{1}(cos(μ_{1}s)e_{1} + sin(μ_{1}s)e_{5}) + y_{2}(cos(μ_{2}s)e_{2}
+ sin(μ_{2}s)e_{6}) + y_{3}(cos(μ_{3}s)e_{3} + sin(μ_{3}s)e_{7}),
(1.1)  μ_{1} + μ_{2} + μ_{3} = 0, μ_{1}^{2} + μ_{2}^{2} + μ_{3}^{2} ≠ 0,

where e_{1}, …, e_{7} is a standard G_{2} basis of the space ℝ^{7} and y_{1}^{2} + y_{2}^{2} + y_{3}^{2} = 1.

Note that these submanifolds are also ruled by totally geodesic spheres S^{2}(1). Here, we prove

Theorem 1.1. Let M be a proper three dimensional CR submanifold of S^{6}(1) ruled by S^{2}(1). If | cosϕ| is the length of the projection of the unit normal to the leaf of ruling at a point, on the almost complex distribution, then ϕ is constant. Moreover M is locally congruent to the immersion:

a) for cosϕ ≠ 0

p(x_{1}, x_{2}, x_{3}) = sin(x_{2} + h)γ × γ′
+ cos(x_{2} + h)(cosx_{1}σ + sinx_{1}(cosϕγ′− sinϕγ × γ′) × σ),

where γ is a curve in the sphere S^{6}(1), with arc length parameter x_{3}, such that ⟨γ × γ′, γ′′⟩ = 0, σ is a curve in S^{6}(1) parameterized by x_{3} orthogonal to γ, γ′, γ × γ′ such that

⟨σ′, γ × γ′⟩ = ⟨σ × σ′, γ⟩ = 0, ⟨σ × σ′, sinϕγ′+ cosϕγ × γ′⟩ = \frac{1}{2} cosϕ,

⟨σ, cosϕ(γ × γ′) × γ′′+ sinϕγ′× γ′′⟩ = 0,

and h is a function of x_{3} such that cos(x_{2} + h) > 0.

b) for cosϕ = 0

f(x_{1}, x_{2}, x_{3}) = cosx_{1} cosx_{2}γ + sinx_{1} cosx_{2}A_{2} + sinx_{2}γ × A_{2},

where γ is a non constant curve in S^{6}(1) parameterized by x_{3} and A_{2} a vector field along γ orthogonal to γ, γ′ and γ × γ′.

3-DIMENSIONAL CR SUBMANIFOLDS OF S^{6} (1) 27

Remark 1.1. Note that immersion (1.1) is of the second type. At least one of the μ_{i} is different from zero, so we can assume that μ_{1} ≠ 0. Then we can parameterise the sphere so that y_{1} = cosx_{1} cosx_{2}, y_{2} = sinx_{1} cosx_{2}, y_{3} = sinx_{2} and take γ to be the curve s ↦ cos(μ_{1}s)e_{1}+sin(μ_{1}s)e_{5} with A_{2} = cos(μ_{2}s)e_{2}+sin(μ_{2}s)e_{6} to obtain the immersion (1.1).

Remark 1.2. In Lemma 2.2 we prove that a three dimensional, totally real submanifold of S^{6}(1) ruled by S^{2}(1) is totally geodesic.

2. Preliminaries

Here, we give a short exposition of how the standard nearly Kähler structure on S^{6}(1) arises from the multiplication of the octonions O.

A vector cross product × of the purely imaginary octonions ImO = ℝ^{7} is given by u × v = \frac{1}{2}(uv − vu). This cross product has many properties similar to those of the cross product in the space ℝ^{3}. In particular, if ⟨, ⟩ denotes the standard inner product of the space ℝ^{7} we have that, see [5],

u × (v × w) + (u × v) × w = 2⟨u, w⟩v − ⟨u, v⟩w − ⟨w, v⟩u, (2.1)

⟨u × v, u × w⟩ = ⟨u, u⟩⟨v, w⟩ − ⟨u, v⟩⟨u, w⟩. (2.2)

An ordered orthonormal basis, respectively moving frame e_{1}, …, e_{7} is said to be a G_{2}-basis, respectively frame, if

e_{3} = e_{1} × e_{2}, e_{5} = e_{1} × e_{4}, e_{6} = e_{2} × e_{4}, e_{7} = e_{3} × e_{4}.

The unit orthogonal vector fields e_{1} and e_{2} and further e_{4} orthogonal to e_{1}, e_{2} and e_{1} × e_{2} determine a G_{2}-frame uniquely. We also give the multiplication table for the cross product.

\begin{matrix}× & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & e_{3} & −e_{2} & e_{5} & −e_{4} & −e_{7} & e_{6} \\ e_{2} & −e_{3} & 0 & e_{1} & e_{6} & e_{7} & −e_{4} & −e_{5} \\ e_{3} & e_{2} & −e_{1} & 0 & e_{7} & −e_{6} & e_{5} & −e_{4} \\ e_{4} & −e_{5} & −e_{6} & −e_{7} & 0 & e_{1} & e_{2} & e_{3} \\ e_{5} & e_{4} & −e_{7} & e_{6} & −e_{1} & 0 & −e_{3} & e_{2} \\ e_{6} & e_{7} & e_{4} & −e_{5} & −e_{2} & e_{3} & 0 & −e_{1} \\ e_{7} & −e_{6} & e_{5} & e_{4} & −e_{3} & −e_{2} & e_{1} & 0 \end{matrix}

The standard nearly Kähler structure on S^{6}(1) is then obtained as follows:

JX = p × X, X ∈ T_{p}S^{6}(1), p ∈ S^{6}(1).

It is clear that J is an orthogonal almost complex structure on S^{6}(1). Also, straightforwardly we have that the group G_{2} of automorphisms of O is precisely the group of isometries of ℝ^{7} preserving the vector cross product and the almost complex structure of the sphere.

If we denote by ⟨ , ⟩, D̄ and D̃ metric and Levi Civita connections of M and M̃, respectively, and by D^{⊥} the corresponding normal connection of the immersion M → M̃ then the formulas of Gauss and Weingarten are given respectively by

D̃_{X} Y = D̄_{X} Y + h(X, Y), D̃_{X} ξ = −A_{ξ}X + D_{X}^{⊥} ξ,

where X and Y are vector fields on M and ξ is a normal vector field on M, and h and A are the second fundamental form and the shape operator, respectively. The second fundamental form and the shape operator are related by

⟨h(X, Y), ξ⟩ = ⟨A_{ξ}X, Y ⟩

Let us denote by ∇, ˜∇ and D the Levi–Civita connections of M, S^{6}(1) and ℝ^{7}. Let h and h̃ be the second fundamental forms corresponding to the immersions M → S^{6}(1) and S^{6}(1) → ℝ^{7}, respectively. If we denote by p the position vector field of the immersion of M into ℝ^{7}, we have h̃(X, Y) = −⟨X, Y ⟩p, and D_{X} p = X, where X, Y ∈ T M. Further, the Gauss and Codazzi equations imply that for X, Y ∈ T M and ξ ∈ T^{⊥} M, ξ ∈ T S^{6}(1) it holds

D_{X} Y = ˜∇_{X} Y + h̃(X, Y) = ∇_{X} Y + h(X, Y) − ⟨X, Y ⟩p,
D_{X} ξ = ˜∇_{X} ξ + h̃(X, ξ) = ˜∇_{X} ξ − ⟨X, ξ⟩p = −A_{ξ}X + ∇_{X}^{⊥} ξ,

where ∇^{⊥} denotes the normal connection corresponding to the immersion of M into S^{6}(1).

Also, if we denote

(∇h)(X, Y, Z) = ∇_{X}^{⊥} h(Y, Z) − h(∇_{X} Y, Z) − h(Y, ∇_{X} Z),

for X, Y, Z ∈ T (M), then the Gauss, Codazzi and Ricci equations yield

R(X, Y, Z, W)= ⟨X, W ⟩⟨Y, Z⟩ − ⟨X, Z⟩⟨Y, W ⟩
+ ⟨h(X, W), h(Y, Z)⟩ − ⟨h(X, Z), h(Y, W)⟩,
(∇h)(X, Y, Z) = (∇h)(Y, X, Z),
⟨R^{⊥} (X, Y)ξ, μ⟩= ⟨[A_{ξ}, A_{μ}]X, Y ⟩.

Also, the following lemma holds straightforwardly.

Lemma 2.1. D_{X} (Y × Z) = D_{X} Y × Z + Y × D_{X} Z.

Lemma 2.2. Let M be a totally real, three dimensional submanifold of S^{6}(1), ruled by totally geodesic S^{2}(1). Then M is locally congruent to a totally geodesic sphere S^{3}(1).

Proof. Assume that the two dimensional totally geodesic distribution D is spanned by unit and orthogonal vector fields E_{1}, E_{2}. Then we have that the second fundamental form vanishes on D. Denote by E_{3} the unit, tangent vector field orthogonal to D. Then the vector fields JE_{1}, JE_{2}, JE_{3} span the normal bundle. Since every three dimensional, totally real submanifold of S^{6}(1) is minimal, see [4], we have that h(E_{3}, E_{3}) = 0. Note that, for tangent vector fields X, Y, Z we have

⟨h(X, Y), JZ⟩ = ⟨A_{JZ} X, Y ⟩ = −⟨Jh(Z, X), Y ⟩ = ⟨h(Z, X), Y ⟩

implying that the form ⟨h(X, Y), JZ⟩ is symmetric in all three components. Therefore, h vanishes identically and M is three dimensional, totally geodesic submanifold of S^{6}(1). □

3. Proof of the main theorem

Here we are dealing with proper, three dimensional, CR submanifolds of S^{6}(1). Since the almost complex distribution is even dimensional, and nontrivial, it follows that dimU = 2 and then dimU^{⊥} = 1. We present one of the convenient moving frames to work with and the relations between the connection coefficients in it, for details see [1, 3].

We denote by p the position vector field of the submanifold. Let then E_{1} and E_{2} = p × E_{1} = JE_{1} be the unit vector fields which span the almost complex distribution, and E_{3} the unit vector field which spans the totally real distribution. Straightforwardly, the normal vector fields are obtained by E_{4} = JE_{3}, E_{5} = E_{1}×E_{3} and E_{6} = E_{2} ×E_{3}. Note, that the choice of E_{3} is unique up to a sign while we have a freedom of rotation in the almost complex distibution, which further reflects to a rotation in the Span(E_{5}, E_{6}).

We denote

ω_{ij}^{k} = ⟨D_{E_{i}} E_{j} , E_{k}⟩, h_{ij}^{k} = ⟨D_{E_{i}} E_{j} , E_{k+3} ⟩, β_{ij}^{k} = ⟨D_{E_{i}} E_{j+3} , E_{k+3} ⟩,

for 1 ⩽ i, j, k ⩽ 3. Since the connection is metric and the second fundamental form symmetric, we have

ω_{ij}^{k} = −ω_{ik}^{j} , h_{ij}^{k} = h_{ik}^{j} , β_{ij}^{k} = −β_{ik}^{j} .

By taking in Lemma 2.1 X ∈ {E_{1}, E_{2}, E_{3}} and Y, Z ∈ {p, E_{1}, …, E_{6}}, we get the following lemma, (see [1]).

Lemma 3.1. For the previously defined coefficients the following relations hold

β_{11}^{3} = −h_{13}^{2}, β_{11}^{2} = 1 + h_{13}^{3}, h_{11}^{1} = −ω_{12}^{3}, h_{12}^{1} = ω_{11}^{3}, β_{21}^{3} = 1 − h_{23}^{2},

β_{21}^{2} = h_{23}^{3}, h_{22}^{1} = ω_{21}^{3}, ω_{22}^{3} = −ω_{11}^{3}, β_{31}^{2} = h_{33}^{3}, β_{31}^{3} = −h_{33}^{2},
h_{23}^{1} = ω_{31}^{3}, h_{13}^{1} = −ω_{32}^{3}, h_{11}^{3} = h_{12}^{2}, h_{11}^{2} = −h_{12}^{3}, h_{22}^{2} = h_{12}^{3},
h_{22}^{3} = −h_{12}^{2}, h_{23}^{2} = h_{13}^{3} − 1, h_{23}^{3} = −h_{13}^{2}, β_{12}^{3} = ω_{11}^{2} − ω_{32}^{3},
β_{22}^{3} = ω_{21}^{2} + ω_{31}^{3}, β_{32}^{3} = ω_{31}^{2} + h_{33}^{1}.

Now we take that M is ruled by S^{2}(1). Then the complementary foliation is spanned by a unit vector field W which can have non vanishing components both in the almost complex and the totally real distribution. Since we still have a freedom of rotating the vector fields E_{1} and E_{2} we can assume that the orthogonal projection of W to U is orthogonal to E_{2}, i.e., W = cosϕE_{1} + sinϕE_{3}, for some differential function ϕ. Then the vector fields V = − sinϕE_{1} +cosϕE_{3} and E_{2} span the totally geodesic distribution. Further the inner products of D_{V}V, D_{V}E_{2}, D_{E_{2}} E_{2} with W, E_{4}, E_{5} and E_{6} vanish. Straightforwardly this yields the following lemma.

Lemma 3.2. Let M be a proper, ruled, three-dimensional CR submanifold of S^{6}(1). Then the following relations hold

h_{11}^{3} = 0, h_{12}^{3} = 0, ω_{21}^{3} = 0, E_{2}(ϕ) = 0, −ω_{21}^{2} cosϕ − h_{12}^{1} sinϕ = 0,
h_{23}^{1} cosϕ − h_{12}^{1} sinϕ = 0, (−1 + h_{13}^{3}) cosϕ = 0, h_{13}^{2} cosϕ = 0,
− cosϕE_{3}(ϕ) + sinϕE_{1}(ϕ) = 0, h_{33}^{1} cos^{2} ϕ − ω_{12}^{3} sin^{2} ϕ + ω_{32}^{3} sin(2ϕ) = 0,
cosϕ(h_{33}^{2} cosϕ − 2h_{13}^{2} sinϕ) = 0, cosϕ(h_{33}^{3} cosϕ − 2h_{13}^{3} sinϕ) = 0.

Note, also, that the Codazzi equation 0 = R(E_{1}, E_{2}, E_{1}, E_{5}) = 2(h_{12}^{1} − ω_{12}^{3}h_{13}^{2}) yields h_{12}^{1} = ω_{12}^{3}h_{13}^{2}. However, now we have to consider separately cases cosϕ ≠ 0 and cosϕ = 0.

3.1. Case cos ϕ ≠ 0 . By taking cosϕ ≠ 0, the relations from Lemma 3.2 straightforwardly reduce to

ω_{21}^{2} = −h_{12}^{1} tanϕ, h_{23}^{1} = h_{12}^{1} tanϕ, h_{13}^{2} = 0, h_{13}^{3} = 1, h_{33}^{2} = 0,

h_{33}^{3} = 2 tanϕ, h_{33}^{1} = tanϕ(−2ω_{32}^{3} + ω_{12}^{3} tanϕ), E_{3}(ϕ) = tanϕE_{1}(ϕ).

Lemma 3.3. We have

ω_{32}^{3} = −ω_{11}^{2} + 2ω_{12}^{3} tanϕ, E_{2}(ω_{11}^{2}) = 1 + (ω_{11}^{2})^{2} − ω_{12}^{3}ω_{31}^{2},
E_{2}(ω_{12}^{3}) = −2ω_{12}^{3}(−ω_{11}^{2} + ω_{12}^{3} tanϕ), E_{1}(ϕ) = 0,
ω_{31}^{2} = ω_{12}^{3} tan^{2} ϕ, E_{3}(ω_{11}^{2}) = E_{1}(ω_{12}^{3}) tan^{2} ϕ,
E_{1}(ω_{11}^{2}) = 2E_{1}(ω_{12}^{3}) tan^{2} ϕ − E_{3}(ω_{12}^{3}).

Proof. Direct computation gives the following equations of Codazzi and Gauss, from which we derive the proclaim.

R(E_{1}, E_{2}, E_{3}, E_{6}) = ω_{11}^{2} + ω_{32}^{3} − 2ω_{12}^{3} tanϕ = 0,
R(E_{1}, E_{2}, E_{1}, E_{2}) = 1 + (ω_{11}^{2})^{2} − ω_{12}^{3}ω_{31}^{2} − E_{2}(ω_{11}^{2}) = 0,
R(E_{1}, E_{2}, E_{1}, E_{4}) = E_{2}(ω_{12}^{3}) + 2ω_{12}^{3}(−ω_{11}^{2} + ω_{12}^{3} tanϕ) = 0,
R(E_{1}, E_{3}, E_{3}, E_{5}) = ω_{31}^{2} − ω_{12}^{3} tan^{2} ϕ = 0,
R(E_{1}, E_{3}, E_{3}, E_{6}) = 2E_{1}(ϕ) sec^{2} ϕ = 0,
R(E_{1}, E_{3}, E_{1}, E_{2}) = −E_{3}(ω_{11}^{2}) + E_{1}(ω_{12}^{3}) tan^{2} ϕ = 0,
R(E_{1}, E_{3}, E_{1}, E_{4}) = E_{1}(ω_{11}^{2}) + E_{3}(ω_{12}^{3}) − 2E_{1}(ω_{12}^{3}) tanϕ = 0.

Direct computation shows that the other Gauss, Codazzi and Ricci equations do not yield any new relations. □

Note that we now have ϕ = const. Also we have

h(E_{1}, E_{1}) = −ω_{12}^{3}E_{4}, h(E_{1}, E_{2}) = 0, h(E_{2}, E_{2}) = 0,
h(E_{3}, E_{3}) = tanϕ(2ω_{11}^{2} − 3ω_{12}^{3} tanϕE_{4} + 2E_{6})

so in a general case distributions U and U^{⊥} are not totally geodesic. We also note that [ρV, E_{2}] = 0, for ρ = (1 + (ω_{11}^{2} − ω_{12}^{3} tanϕ)^{2})^{−1/2} . Moreover, we have

ρV (ω_{11}^{2}) = \frac{sinϕ(E_{3}(ω_{12}^{3}) − E_{1}(ω_{12}^{3}) tanϕ)}{\sqrt{1 + (ω_{11}^{2} − ω_{12}^{3} tanϕ)^{2}}} ,

ρV (ω_{12}^{3}) = \frac{cosϕE_{3}(ω_{12}^{3}) − sinϕE_{1}(ω_{12}^{3})}{\sqrt{1 + (ω_{11}^{2} − ω_{12}^{3} tanϕ)^{2}}} ,

(3.1) ρV (ω_{11}^{2} − ω_{12}^{3} tanϕ) = 0, E_{2}(ω_{11}^{2} − ω_{12}^{3} tanϕ) = 1 + (ω_{11}^{2} − ω_{12}^{3} tanϕ)^{2}.

We give further a construction of the immersion. Recall that [ρV, E_{2}] = 0. Therefore, there exist local coordinates x_{1}, x_{2}, x_{3} in the neighborhood of a point p ∈ M such that ρV = ∂_{x_{1}} , E_{2} = ∂_{x_{2}} . Here, the choice of the coordinate x_{3} is essentially arbitrary. Moreover, (3.1) implies that

∂_{x_{1}} (ω_{11}^{2} − ω_{12}^{3} tanϕ) = 0, ∂_{x_{2}} (ω_{11}^{2} − ω_{12}^{3} tanϕ) = 1 + (ω_{11}^{2} − ω_{12}^{3} tanϕ)^{2}.

Hence, we have ω_{11}^{2} − ω_{12}^{3} tanϕ = tan(x_{2} + h(x_{3})), where the differentiable function h is independent of x_{1}, x_{2}. Also, we have a freedom of choosing h up to an addend π, so we can assume that cos(x_{2} + h) > 0. Further we have E_{2}(ω_{12}^{3}) = ∂_{x_{2}} ω_{12}^{3} = 2ω_{12}^{3} tan(x_{2} + h). Therefore there exists a differentiable function d depending on x_{1} and x_{3} such that

ω_{12}^{3} = d sec^{2}(x_{2} + h), and further ω_{11}^{2} = d sec^{2}(x_{2} + h) tanϕ + tan(x_{2} + h).

Now, straightforward computation shows that

D_{∂_{x_{1}}} ∂_{x_{1}} = ∂_{x_{1} x_{1}}^{2} p = − cos^{2}(x_{2} + h)p + cos(x_{2} + h) sin(x_{2} + h)∂_{x_{2}} p,

D_{∂_{x_{1}}} ∂_{x_{2}} = ∂_{x_{1} x_{2}}^{2} p = − tan(x_{2} + h)∂_{x_{1}} p,

D_{∂_{x_{2}}} ∂_{x_{2}} = ∂_{x_{2} x_{2}}^{2} p = −p. (3.2)

The last equation yields p = cosx_{2}A + sinx_{2}B for some vector fields A and B depending on x_{1} and x_{3}. Since p is unit, straightforwardly we get that A and B are unit and mutually orthogonal. Then, the first equation of (3.2) reduces to

cosx_{2}(∂_{x_{1} x_{1}}^{2} A + cosh(A cosh − B sinh))

+ sinx_{2}(∂_{x_{1} x_{1}}^{2} B − sinh(A cosh − B sinh)) = 0,

which further implies

∂_{x_{1} x_{1}}^{2} A + cosh(A cosh − B sinh) = 0,
∂_{x_{1} x_{1}}^{2} B − sinh(A cosh − B sinh) = 0.

Further, it follows ∂_{x_{1} x_{1}}^{2} (A cosh − B sinh) = −(A cosh − B sinh), so we can put

(3.3)  A cosh − B sinh = cosx_{1}P + sinx_{1}Q,

for some vector fields P and Q depending only on x_{3}. Similarly as before P and Q are unit and orthogonal. Finally, the second equation of (3.2) simplifies to sec(h + x_{2})(cosh ∂_{x_{1}} B + sinh ∂_{x_{1}} A) = 0. Integration over x_{1} then gives sinhA + coshB = G, where the unit vector field G depends only on x_{3}, and is orthogonal to vector field (3.3), for arbitrary x_{1}. Therefore G is orthogonal to both P and Q. Using the last equation along with (3.3) we get

A = cosh(cosx_{1}P + sinx_{1}Q) + sinhG,
B = − sinh(cosx_{1}P + sinx_{1}Q) + coshG.

Now, the immersion is given by

p(x_{1}, x_{2}, x_{3}) = cos(x_{2} + h)(cosx_{1}P + sinx_{1}Q) + sin(x_{2} + h)G,

and the coordinate vector fields by

∂_{x_{1}} p = cos(x_{2} + h)(− sinx_{1}P + cosx_{1}Q) = cos(x_{2} + h)(− sinϕE_{1} + cosϕE_{3}),
∂_{x_{2}} p = − sin(x_{2} + h)(cosx_{1}P + sinx_{1}Q) + cos(x_{2} + h)G = E_{2},
∂_{x_{3}} p = cos(x_{2} + h)(∂_{x_{3}} hG + cosx_{1}∂_{x_{3}} P + sinx_{1}∂_{x_{3}} Q)
+ sin(x_{2} + h)(∂_{x_{3}} G − ∂_{x_{3}} h(cosx_{1}P + sinx_{1}Q)).

Then E_{1} = E_{2} × p = G × (cosx_{1}P + sinx_{1}Q), and further

− sinϕ = ⟨− sinx_{1}P + cosx_{1}Q, E_{1}⟩
= cos^{2} x_{1}⟨Q, G × P ⟩ − sin^{2} x_{1}⟨P, G × Q⟩ = ⟨P × Q, G⟩.

It follows that

⟨G + sinϕP × Q, P × Q⟩ = 0, G + sinϕP × Q ⊥ P, Q,
⟨G + sinϕP × Q, G + sinϕP × Q⟩ = cos^{2} ϕ ≠ 0.

Since we can assume that cosϕ > 0 (we can change the sign of W) the following lemma holds.

Lemma 3.4. The vector fields P, Q and T = secϕ(G + sinϕP × Q) determine a G_{2}-frame.

Therefore, we can denote e_{1} = P, e_{2} = Q, e_{4} = T, and other e_{i} accordingly to the relations in a G_{2} frame. Here we have G = cosϕe_{4} − sinϕe_{3}. Moreover, it follows that

E_{3} = secϕ(cosx_{1}Q − sinx_{1}P + sinϕE_{1})
= − sinx_{1}(cosϕe_{1} + sinϕe_{5}) + cosx_{1}(cosϕe_{2} − sinϕe_{2}),
E_{4} = p × E_{3} = cos(h + x_{2})(cosϕe_{3} + sinϕe_{4}) + sin(h + x_{2})(sinx_{1}e_{5} − cosx_{1}e_{6}),
E_{5} = E_{1} × E_{3} = e_{7},
E_{6} = − sin(h + x_{2})(cosϕe_{3} + sinϕe_{4}) + cos(h + x_{2})(sinx_{1}e_{5} − cosx_{1}e_{6}).

Denote by z_{ij} = ⟨∂_{x_{3}} e_{i}, e_{j} ⟩, the differentiable functions of x_{3}. Since the connection is metrical, we have z_{ji} = −z_{ij} .

Lemma 3.5. It holds

\begin{matrix}z_{34} = z_{16} − z_{25}, & z_{35} = z_{17} + z_{24}, & z_{36} = −z_{14} + z_{27}, & z_{37} = −z_{15} − z_{26}, \\ z_{56} = z_{12} + z_{47}, & z_{57} = z_{13} − z_{46}, & z_{67} = z_{23} + z_{45}. \end{matrix}

Proof. From Lemma 2.1 we have ∂_{x_{3}} (e_{i} ×e_{j}) = ∂_{x_{3}} e_{i} ×e_{j} +e_{i} ×∂_{x_{3}} e_{j}. Taking

i, j ∈ {1, …, 7}, we obtain the assertion. □

The previous lemma obviously holds for an arbitrary G_{2}-frame. Let us now consider this particular one.

Lemma 3.6. Coefficients z_{ij} satisfy

z_{26} = z_{15} ≠ 0, z_{16} = 0, z_{17} = 0, z_{25} = 0, z_{27} = 0,

z_{46} = −z_{13} − 2z_{14} tanϕ, z_{45} = z_{23} + 2z_{24} tanϕ, z_{13} = −z_{14} tanϕ,

z_{23} = −z_{24} tanϕ, z_{47} = −2z_{15} tanϕ.

Proof. For p to be the CR immersion of the needed type we have to impose the following conditions. The vector field ∂_{x_{3}} p has to be independent of ∂_{x_{1}} p and ∂_{x_{2}} p, and orthogonal to the vector fields E_{4}, E_{5} and E_{6}. We have

0 = ⟨∂_{x_{3}} p, E_{5}⟩ = cos(h + x_{2})(z_{17} cosx_{1} + z_{27} sinx_{1})
+ sin(h + x_{2})(z_{47} cosϕ + (z_{15} + z_{26}) sinϕ)).

Since cosx_{1} and sinx_{1} are the only functions of x_{1} in this relation, we get that z_{17} = z_{27} = 0 and z_{47} cosϕ + (z_{15} + z_{26}) sinϕ = 0. The relation z_{15} = z_{26} and other ones are obtained similarly, taking in the expressions for ⟨∂_{x_{3}} p, E_{4}⟩ and ⟨∂_{x_{3}} p, E_{6}⟩ the coefficients multiplying independent functions of x_{1}. Finally, if we denote by ∂_{x_{3}} p^{pr} the projection of ∂_{x_{3}} p to Span(∂_{x_{1}} p, ∂_{x_{2}} p) we get

∂_{x_{3}} p − ∂_{x_{3}} p^{pr} = z_{15} cos(h + x_{2})(cosx_{1}e_{5} + sinx_{1}e_{6}) ≠ 0

which finishes the proof. □

Note that now it follows that e′_{7} = 2z_{15}(e_{3} +tanϕe_{4}) ≠ 0. By possible rescaling of the coordinate x_{3}, we can assume that ∥e′_{7}∥ = 1, i.e., that the sphere curve γ(x_{3}) = e_{7}(x_{3}) is parameterized by arc length and that 2z_{15} = cosϕ. Then

γ × γ′ = e_{7} × (cosϕe_{3} + sinϕe_{4}) = − sinϕe_{3} + cosϕe_{4} = G,

and further

e_{3} = cosϕγ′− sinϕγ × γ′, e_{4} = sinϕγ′+ cosϕγ × γ′.

Since

(3.4)  e′_{1} = z_{12}e_{2} + z_{14}(− tanϕe_{3} + e_{4}) + z_{15}e_{5} ≠ 0,
(3.5)  e′_{2} = −z_{12}e_{1} + z_{24}(− tanϕe_{3} + e_{4}) + z_{15}e_{6}

we also have that σ = e_{1} is a nonconstant sphere curve, not necessarily parameterized by its arc length. Since e_{1}, e_{3} and e_{4} determine a G_{2}-frame, we have that σ is orthogonal to γ, γ′ and γ × γ′. From (3.4) we deduce that σ′ is orthogonal to G = γ × γ′, e_{7} = γ and e_{6} = e_{1} × e_{7} = σ × γ which is, by (2.2), equivalent to ⟨σ′× σ, γ⟩ = 0. Also, (3.4) implies

⟨σ′, σ × e_{4}⟩ = ⟨σ′× σ, sinϕγ′+ cosϕγ × γ′⟩ = ⟨e′_{1}, e_{5}⟩ = \frac{1}{2} cosϕ = const.

Similarly, (3.5) implies that (e_{3} × σ)′ is orthogonal to G and e_{5} = e_{1} × e_{4}. We note that ⟨(e_{3} × e_{1})′, e_{1} × e_{4}⟩ = ⟨e′_{3} × e_{1}, e_{1} × e_{4}⟩ + ⟨e_{3} × e′_{1}, e_{1} × e_{4}⟩. By using (2.2) and (2.1) we get

⟨e_{3} × e′_{1}, e_{1} × e_{4}⟩ = −⟨e′_{1}, e_{3} × (e_{1} × e_{4})⟩ = ⟨e′_{1}, e_{6}⟩ = ⟨e′_{1} × e_{1}, γ⟩ = 0.

Therefore, we have 0 = ⟨e′_{3} × e_{1}, e_{1} × e_{4}⟩ = ⟨−e′_{3}, e_{4}⟩ and further

0 = ⟨e′_{3}, e_{4}⟩ = ⟨cosϕγ′′− sinϕ(γ × γ′)′, sinϕγ′+ cosϕ(γ × γ′)⟩
= cos^{2} ϕ⟨γ′′, γ × γ′⟩ − sin^{2} ϕ⟨γ × γ′′, γ′⟩ = −⟨γ′′, γ × γ′⟩.

Here we use the fact that the sphere curves γ′ and γ × γ′ are orthogonal to their tangent vector fields γ′′and (γ × γ′)′, respectively.

Let us investigate the orthogonality condition for (e_{3} × σ)′ and G. We have

⟨e_{3} × e′_{1}, γ × γ′⟩ = −⟨e′_{1}, e_{3} × (γ × γ′)⟩
= −⟨e′_{1}, cosϕγ′× (γ × γ′)⟩ = − cosϕ⟨e′_{1}, γ⟩ = 0,

so we are left with

(3.6)  0 = ⟨e′_{3} × e_{1}, γ × γ′⟩ = ⟨e′_{3}, e_{1} × (γ × γ′)⟩ = ⟨e_{1}, (γ × γ′) × e′_{3}⟩
= ⟨e_{1}, cosϕ(γ × γ′) × γ′′− sinϕ(γ × γ′) × (γ × γ′′)⟩.

Since (2.1) implies

(γ × γ′) × (γ × γ′′) = γ′× (γ × (γ × γ′′))
= γ′× (⟨γ, γ′′⟩γ − ⟨γ, γ⟩γ′′) = −γ′× (γ + γ′′)

and since σ and γ × γ′ are orthogonal, relation (3.6) becomes

⟨σ, cosϕ(γ × γ′) × γ′′+ sinϕγ′× γ′′⟩ = 0.

Straightforward computation shows that taking curves γ and σ that satisfy the listed conditions satisfy the relations of Lemma 3.6 and, moreover that we obtain a CR submanifold of the required form.

3.2. Case cos ϕ = 0 . In this case we have W = E_{3} and the submanifold is foliated by almost complex spheres. We will now present a construction of the submanifold following the method given in [1]. If p ∈ M is a point of the submanifold, then there exists a G_{2} basis e_{1}, …, e_{7} of the space ℝ^{7} such that e_{1} = p, e_{4} = E_{3}(p) and the tangent space of the totally geodesic leaf at the point p is spanned by e_{2} and e_{3} = p × e_{2}. We can parameterize that leaf by

(cosx_{1} cosx_{2}, sinx_{1} cosx_{2}, sinx_{2}, 0, 0, 0, 0)

for x_{1}, x_{2} in some neighborhood of (0, 0). We denote by γ the integral curve for the vector field E_{3}, parameterized by x_{3} such that γ(0) = p. If γ(x_{3}) is a point of the curve, then there also exists a G_{2} transformation A(x_{3}) mapping, respectively, p into γ(x_{3}), and the vectors E_{1}(p) and E_{3}(p) into E_{1}(γ(x_{3})) and E_{3}(γ(x_{3})). Note that we have a possibility of choosing the vector field E_{1} belonging to the almost complex distribution. Now, we have that locally the immersion is given by f(x_{1}, x_{2}, x_{3}) = A(x_{3})(cosx_{1} cosx_{2}, sinx_{1} cosx_{2}, sinx_{2})^{t}, for a differentiable G_{2} matrix function A(x_{3}). Denoting by A_{i} the columns of A, we obtain

f(x_{1}, x_{2}, x_{3}) = cosx_{1} cosx_{2}A_{1}(x_{3}) + sinx_{1} cosx_{2}A_{2}(x_{3}) + sinx_{2}A_{3}(x_{3}).

Since the point p, obtained for x_{1} = x_{2} = 0, is mapped into γ(x_{3}), we have that A_{1} = γ. Similarly e_{4} is mapped into γ′, so e_{2} is mapped into A_{2} orthogonal to γ, γ′ and γ × γ′. Moreover we have A_{3} = A_{1} × A_{2} = γ × A_{2}. Therefore, γ, A_{2} and γ′ determine a G_{2} frame.

We have

∂_{x_{1}} f = − sinx_{1} cosx_{2}A_{1} + cosx_{1} cosx_{2}A_{2},
∂_{x_{2}} f = − sinx_{2} cosx_{1}A_{1} − sinx_{2} sinx_{1}A_{2} + cosx_{2}A_{3},

and further

∂_{x_{1} x_{1}}^{2} f = − cosx_{2}(cosx_{1}A_{1} + sinx_{1}A_{2}) = − cosx_{2}^{2}f + sinx_{2} cosx_{2}∂_{x_{2}} f,
∂_{x_{1} x_{2}}^{2} f = sinx_{2}(sinx_{1}A_{1} − cosx_{1}A_{2}) = − tanx_{2}∂_{x_{1}} f,
∂_{x_{2} x_{2}}^{2} f = −p.

Also, straightforwardly we have f × \frac{∂_{x_{1}} f}{cos x_{2}} = ∂_{x_{2}} f, so ∂_{x_{1}} f and ∂_{x_{2}} f span an almost complex distribution, with integral manifolds being totally geodesic spheres, parameterized by x_{1}, x_{2}. Moreover this makes f a CR immersion of a required form. Also, note that by reparameterization of the curve γ we obtain the same CR submanifold, so the condition that x_{3} is the arc-length parameter is not necessary. This completes the proof.

Acknowledgments. Supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, project 174012.

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Faculty of Mathematics (Received 31 03 2016) University of Belgrade Belgrade Serbia

    mira@math.rs