Minimal submanifolds in $\textrm{Sol}_1^4$

Abstract

In this paper minimal invariant, totally real and CR-submanifolds in the 4-dimensional homogeneous solvable Lie group $\textrm{Sol}_1^4$ equipped with standard globally conformal Kähler structure are studied. It is proved that the only minimal invariant surfaces of $\textrm{Sol}_1^4$ are totally geodesic hyperbolic planes. The minimal totally real submanifolds with tangent or normal Lee vector field are classified. Finally, CR-submanifolds with tangent and normal Lee vector field are examined. Three types of minimal proper CR-submanifolds with tangent Lee vector field are discovered and the statement that only minimal CR-submanifold normal to the Lee field is nilradical $\textrm{Nil}_3$ is proved.

A The solvable Lie group G(k, n)

1.1 A.1

Let us consider the solvable Lie group

$$\begin{aligned} G(k)=\left\{ (x,y,z):=\left. \left( \begin{array}{ccc} e^{kz} &{} 0 &{}x\\ 0 &{} e^{-kz} &{}y\\ 0 &{} 0 &{}1 \end{array}\right) : x,y,z\in {\mathbb {R}} \right\} \right. , \end{aligned}$$

where $k\in {\mathbb {R}}^{\times }$. The Lie algebra $\mathfrak {g}(k)$ of G(k) is given by

$$\begin{aligned} \mathfrak {g}(k)=\left\{ (u,v,w):=\left. \left( \begin{array}{ccc} kw &{} 0 &{}u\\ 0 &{} -kw &{}v\\ 0 &{} 0 &{}0 \end{array}\right) : x,y,z\in {\mathbb {R}} \right\} \right. . \end{aligned}$$

We take the following basis of $\mathfrak {g}(k)$:

$$\begin{aligned} \bar{E}_1=\left( \begin{array}{ccc} 0 &{} 0 &{}1\\ 0 &{} 0 &{}0\\ 0 &{} 0 &{}0 \end{array}\right) , \quad \bar{E}_2=\left( \begin{array}{ccc} 0 &{} 0 &{}0\\ 0 &{} 0 &{}1\\ 0 &{} 0 &{}1 \end{array}\right) , \quad \bar{E}_3=\left( \begin{array}{ccc} k &{} 0 &{}0\\ 0 &{} -k &{}0\\ 0 &{} 0 &{}0 \end{array}\right) . \end{aligned}$$

Then the right invariant vector fields determined by this basis are given by

$$\begin{aligned} \bar{E}_1=\frac{\partial }{\partial x}, \quad \bar{E}_2=\frac{\partial }{\partial y}, \quad \bar{E}_3=\frac{\partial }{\partial z} +k\left( x\frac{\partial }{\partial x} -y\frac{\partial }{\partial y}\right) . \end{aligned}$$

The dual 1-forms of $\{\bar{E}_1,\bar{E}_2,\bar{E}_3\}$ are given by

$$\begin{aligned} \bar{\Theta }^1=dx-kxdz, \quad \bar{\Theta }^2=dy+kydz, \quad \bar{\Theta }^3=dz. \end{aligned}$$

We equip a right invariant Riemannian metric g of G(k) by $g=(\bar{\Theta }^1)^2+(\bar{\Theta }^2)^2+(\bar{\Theta }^3)^2$.

Let us choose k so that $2\cosh k\in {\mathbb {Z}}\smallsetminus \{2\}$, then there exits a discret subgroup $\varGamma (k)$ of G(k) so that the quotient $M^{3}(k):=G(k)/\varGamma (k)$ is a compact 3-manifold. Moreover the right invariant 1-forms $\bar{\Theta }^1$, $\bar{\Theta }^2$ and $\bar{\Theta }^3$ descend to 1-forms on $M^{3}(k)$. For simplicity of notation we denote the induced 1-forms on $M^{3}(k)$ by the same letters $\bar{\Theta }^1$, $\bar{\Theta }^2$ and $\bar{\Theta }^3$. Hence the Riemannian metric g also descends to $M^{3}(k)$. The real cohomology groups of $M^{3}(k)$ are computed as

$$\begin{aligned}{} & {} \textrm{H}^{0}(M^{3}(k);{\mathbb {R}})=\{1\}, \quad \textrm{H}^{1}(M^{3}(k);{\mathbb {R}})=\{[\bar{\Theta }^3]\}, \\{} & {} \textrm{H}^{2}(M^{3}(k);{\mathbb {R}})=\{[\bar{\Theta }^1\wedge \bar{\Theta }^2]\}, \quad \textrm{H}^{3}(M^{3}(k);{\mathbb {R}}) =\{[\bar{\Theta }^1\wedge \bar{\Theta }^2\wedge \bar{\Theta }^3]\}. \end{aligned}$$

Thus the Betti numbers are given by

$$\begin{aligned} b_{1}(M^{3}(k))=b_{2}(M^3(k))=b_3(M^3(k))=1. \end{aligned}$$

1.2 A.2

In 1989, de Andrés, Cordero, Fernández and Mencía [10] gave an interesting family of 4-dimensional solvmanifolds equipped with LCK structure. Let G(k, n) be the connected solvable Lie group consisting of matrices of the form

$$\begin{aligned} (x,y,z,t):=\left( \begin{array}{ccccc} e^{kz} &{} 0 &{} 0 &{} 0 &{}x\\ -nye^{kz} &{} 1 &{} 0 &{} 0 &{} t\\ 0 &{} 0 &{} e^{-kz} &{} 0 &{} y\\ 0 &{} 0 &{} 0 &{} 1 &{} z\\ 0 &{} 0 &{} 0 &{} 0 &{}1 \end{array}\right) , \end{aligned}$$

where k, $n\in {\mathbb {R}}^{\times }$.

The inverse element of (x, y, z, t) is given by

$$\begin{aligned} (x,y,z,t)^{-1}=\left( \begin{array}{ccccc} e^{-kz} &{} 0 &{} 0 &{} 0 &{} -xe^{-kz}\\ ny &{} 1 &{} 0 &{} 0 &{} -nxy-t\\ 0 &{} 0 &{} e^{kz} &{} 0 &{} -ye^{kz}\\ 0 &{} 0 &{} 0 &{} 1 &{} -z\\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{array}\right) . \end{aligned}$$

The right Maurer–Cartan form of G(k, n) is computed as

$$\begin{aligned}&d(x,y,z,t)\,(x,y,z,t)^{-1}\\&\quad =\begin{pmatrix} ke^{kz}dz &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad dx \\ -ne^{kz}(dy+kydz) &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad dt\\ 0 &{}\quad 0 &{}\quad -ke^{-kz}dz &{}\quad 0 &{}\quad dy\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad dz\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \end{pmatrix} \left( \begin{array}{ccccc} e^{-kz} &{} 0 &{} 0 &{} 0 &{} -xe^{-kz}\\ ny &{} 1 &{} 0 &{} 0 &{} -nxy-t\\ 0 &{} 0 &{} e^{kz} &{} 0 &{} -ye^{kz}\\ 0 &{} 0 &{} 0 &{} 1 &{} -z\\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{array}\right) \\&\quad = \Theta ^1 E_1+\Theta ^2E_2+\Theta ^3 E_3+\Theta ^4E_4, \end{aligned}$$

where

$$\begin{aligned} E_1= & {} \left( \begin{array}{ccccc} 0 &{} 0 &{} 0 &{} 0 &{} 1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 \end{array}\right) , \quad E_2=\left( \begin{array}{ccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ -n &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 \end{array}\right) , \\ E_3= & {} \left( \begin{array}{ccccc} k &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} -k &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 \end{array}\right) , \quad E_4= \left( \begin{array}{ccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 \end{array}\right) . \end{aligned}$$

and

$$\begin{aligned} \Theta ^1=dx-kx dz, \quad \Theta ^2=dy+kydz, \quad \Theta ^3=dz, \quad \Theta ^4=dt+nx(dy+kydz). \end{aligned}$$

Then $\{E_1,E_2,E_3,E_4\}$ is a basis of the Lie algebra $\mathfrak {g}(k,n)$ of G(k, n). We denote the right invariant vector fields obtained from $E_1$, $E_2$, $E_3$ and $E_4$ by right translations by the same letter. Then we get

$$\begin{aligned} E_1=\frac{\partial }{\partial x},\quad E_2=\frac{\partial }{\partial y}-nx\frac{\partial }{\partial t}, \quad E_3=\frac{\partial }{\partial z}+kx \frac{\partial }{\partial x}-ky\frac{\partial }{\partial y}, \quad E_4=\frac{\partial }{\partial t}. \end{aligned}$$

The Riemannian metric $g=(\Theta ^1)^2+(\Theta ^2)^2+(\Theta ^3)^2+ (\Theta ^4)^2$ is a right invariant Riemannian metric on G(k, n). The right invariant metric g has the components

$$\begin{aligned} \begin{pmatrix} 1 &{}\quad 0 &{}\quad -kx &{}\quad 0\\ 0 &{}\quad 1+n^2x^2 &{}\quad ky(1+n^2x^2) &{}\quad nx\\ -kx &{}\quad ky(1+n^2x^2) &{}\quad 1+k^2(x^2+y^2+n^2x^2y^2) &{}\quad knxy\\ 0 &{}\quad nx &{}\quad knxy &{}\quad 1 \end{pmatrix} \end{aligned}$$

relative to the coordinates (x, y, z, t).

The Riemannian 4-manifold $({\mathbb {R}}^2(x,y,z,t),g)$ can be seen in a paper [26, §3] by Matsumoto and Pripoae.

The Levi–Civita connection is given by ([10, p. 230]:

$$\begin{aligned} \begin{array}{llll} \nabla _{E_1}E_1=-kE_3, &{}\nabla _{E_1}E_2=-\frac{n}{2}E_4, &{} \nabla _{E_1}E_3=k E_1, &{} \nabla _{E_1}E_4=\frac{n}{2}E_2, \\ \nabla _{E_2}E_1=\frac{n}{2}E_4, &{} \nabla _{E_2}E_2=k E_3, &{} \nabla _{E_2}E_3=-k E_2, &{} \nabla _{E_2}E_4=-\frac{n}{2}E_1, \\ \nabla _{E_3}E_1=0, &{}\nabla _{E_3}E_2=0, &{} \nabla _{E_3}E_3=0, &{} \nabla _{E_3}E_4=0, \\ \nabla _{E_4}E_1=\frac{n}{2}E_2, &{}\nabla _{E_4}E_2=-\frac{n}{2}E_1, &{} \nabla _{E_4}E_3=0 &{} \nabla _{E_4}E_4=0. \end{array} \end{aligned}$$

Then we have

$$\begin{aligned} {[}E_1,E_2]=-n\, E_4, \quad [E_1,E_3]=kE_1, \quad [E_2,E_3]=-kE_2. \end{aligned}$$

1.3 A.3

Let us choose $n=k\in {\mathbb {R}}^{\times }$ and introduce a right invariant g-orthogonal almost complex structure J on G(k, k) by

$$\begin{aligned} JE_1=E_4, \quad JE_2=E_3, \quad JE_3=-E_2, \quad JE_4=-E_1. \end{aligned}$$

Then one can see that (G(k, k), J, g) is a GCK surface. The resulting GCK surface coincides with ${\mathbb {R}}^{4}(k)$ studied in [18, 26, 27, 29]. Note that J has components

$$\begin{aligned} J=\begin{pmatrix} 0 &{}\quad 0 &{}\quad 0 &{}\quad -1\\ 0 &{}\quad -ky &{}\quad -(1+k^2y^2) &{}\quad 0\\ 0 &{}\quad 1 &{}\quad ky &{}\quad 0\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \end{pmatrix} \end{aligned}$$

relative to the coordinated (x, y, z, t).

Kamishima [20] proved G(1, 1) is holomorphically isometric to the model space $\textrm{Sol}^{4}_{1}$ equipped with Tricerri’s LCK structure. As a result the GCK surface ${\mathbb {R}}^4(1)$ studied in [29] is identified with $\textrm{Sol}_1^4$.

1.4 A.4

Next we consider almost complex structures on G(k, n) for $n\not =0$ and $k\not =0$. Let us introduce a right invariant almost complex structure $\bar{J}_{k,n}$ on G(k, n) by

$$\begin{aligned} \bar{J}_{k,n}E_1=\frac{n}{k}\,E_4, \quad \bar{J}_{k,n}E_2=E_3, \quad \bar{J}_{k,n}E_3=-E_2, \quad \bar{J}_{k,n}E_4=-\frac{k}{n}\,E_1. \end{aligned}$$

Relative to the global coordinates (x, y, z, t), $\bar{J}_{k,n}$ has components:

$$\begin{aligned} \bar{J}_{k,n}=\begin{pmatrix} 0 &{}\quad 0 &{}\quad 0 &{}\quad -\frac{k}{n}\\ 0 &{}\quad -ky &{}\quad -(1+k^{2}y^{2}) &{}\quad 0\\ 0 &{}\quad 1&{}\quad ky &{}\quad 0\\ \frac{n}{k} &{}\quad 0 &{}\quad 0 &{}\quad 0 \end{pmatrix}. \end{aligned}$$

The almost complex structure $\bar{J}_{k,n}$ is integrable for arbitrary k and n. However it is g-orthogonal on G(k, n) if and only if $k=-\pm n$.

1.5 A.5

Let us choose $n\in {\mathbb {Z}}\smallsetminus \{0\}$ and $k\in {\mathbb {R}}^{\times }$ so that $2\cosh k\in {\mathbb {Z}}\smallsetminus \{2\}$.

Here we recall the following fundamental theorem due to Kobayashi [21].

Theorem A.1

There is a bijective correspondence between equivalence class of principal circle bundles over a manifold M and the cohomology group $\textrm{H}^2(M;{\mathbb {Z}})$. For a prescribed integral closed 2-form $\Omega$ on M, there exists a principal circle bundle P over M with connection form $\zeta$ whose curvature form is $\Omega$.

Since $\textrm{H}^{2}(M^{3}(k);{\mathbb {R}})= \{[\bar{\Theta }^1\wedge \bar{\Theta }^2]\}$, there exists $\lambda \in {\mathbb {R}}$ such that $\lambda [\bar{\Theta }^1\wedge \bar{\Theta }^2]$ is integral. Hence for each $n\in {\mathbb {Z}}\setminus \{0\}$, there exists a principal circle bundle $M^{4}(k,n)$ over G(k) corresponding to $\lambda [\bar{\Theta }^1\wedge \bar{\Theta }^2] \in \textrm{H}^{2}(M^{3}(k);{\mathbb {Z}})$. The connection form of this bundle has curvature $n\lambda \bar{\Theta }^1\wedge \bar{\Theta }^2$. de Andrés, Cordero, Fernández and Mencía [10] proved that there exists a discrete subgroup $\varGamma (k,n)$ of G(k, n) such that the compact quotient $G(k,n)/\varGamma (k,n)$ is $M^{4}(k,n)$. The Riemannian metric g on G(k, n) and the complex structure $\bar{J}_{k,n}$ descend to the compact quotient $M^4(k,n)$. The resulting structure is an LCK structure on $M^{4}(k,n)$ (For more details, see [10] and [12, pp. 26–27]). The Betti numbers of $M^{4}(k,n)$ are given by [10]:

$$\begin{aligned} b_1(M^{4}(k,n))=1, \quad b_2(M^{4}(k,n))=0, \quad b_3(M^{4}(k,n))=1. \end{aligned}$$

Hence $M^{4}(k,n)$ can not be Kähler.

1.6 A.6

On the other hand, Matsumoto and Pripoae [26] introduced the following almost complex structure (see [26, (3.7)])

$$\begin{aligned} J_{k,n,\lambda }=\begin{pmatrix} 0 &{}\quad k(1-\lambda ^{-1})x&{} \quad k^2(1-\lambda ^{-1})xy &{}\quad -\frac{k}{n}\lambda ^{-1}\\ 0 &{}\quad -ky &{}\quad -(1+k^2y^2) &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad ky &{}\quad 0 \\ \frac{n}{k}\lambda &{}\quad 0 &{}\quad 0 &{}\quad 0 \end{pmatrix},\quad \lambda \in {\mathbb {R}}^{\times } \end{aligned}$$

on G(k, n). To get complex structures, they choose $\lambda$ as $\lambda =k/n$, then $J_{k,n,k/n}$ is

$$\begin{aligned} J_{k,n,k/n}=\begin{pmatrix} 0 &{}\quad (k-n)x&{}\quad k(k-n)xy &{}\quad -1\\ 0 &{}\quad -ky &{}\quad -(1+k^2y^2) &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad ky &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \end{pmatrix}. \end{aligned}$$

This is the correct formula for [26, (3.10)]. It should be remarked that $\bar{J}_{k,n}=J_{k,n,k/n}$ if and only if $n=k$. Moreover Matsumoto and Pripoae [26, Theorem 3.3] showed that $J_{k,n,k/n}$ is integrable if and only if $k=n$.

As a result on G(k, k), all the complex structures J, $\bar{J}_{k,k}$ and $J_{k,k,1}$ are identical.