2. Preliminaries

2.1. Hypersurface geometry

Let $M$ be an immersed hypersurface of a Riemannian manifold $(\widetilde {M},g)$ . Since we will work locally, we may assume that $M$ is orientable. Take a unit normal vector field $\nu$ of $M$ in $\widetilde {M}$ . We denote the Riemannian metric on $M$ induced from $g$ by the same letter $g$ and call it the first fundamental form. We denote by $TM$ and $T^{\perp }M$ the tangent bundle and normal bundle of $M$ , respectively.

Then the Levi-Civita connection $\nabla ^{M}$ of the first fundamental form is described by the Gauss formula:

\begin{equation*} \nabla _{X}Y=\nabla ^{M}_{X}Y+h(X,Y)\nu, \quad X,Y\in \mathfrak {X}(M), \end{equation*}

where

\begin{equation*} \nabla ^{M}_XY=\tan (\nabla _{X}Y), \quad \mathrm {and} \quad h(X,Y)\nu =\mathrm {nor}(\nabla _{X}Y). \end{equation*}

The symmetric tensor field $h$ is called the second fundamental form derived from $\nu$ .

For a vector field $\xi$ normal to $M$ , we have the decomposition (Weingarten formula):

\begin{equation*} \nabla _{X}\xi =-A_{\xi }X+\nabla ^{\perp }_{X}\xi, \quad X\in \mathrm {X}(M), \end{equation*}

where

\begin{equation*} -A_{\xi }X=\tan (\nabla _{X}\xi ), \quad \mathrm {and} \quad \nabla ^{\perp }_{X}\xi =\mathrm {nor}(\nabla _{X}\xi ). \end{equation*}

The Weingarten formula induces a connection $\nabla ^{\perp }$ of the normal bundle $T^{\perp }M$ (called the normal connection). Next, $A_{\xi }$ is an endomorphism field which is called the Weingarten operator corresponding to $\xi$ . In particular, for the unit normal vector field $\nu$ , Weingarten formula is simplified as

\begin{equation*} \nabla _{X}\nu =-AX,\quad X\in \mathfrak {X}(M). \end{equation*}

The endomorphism field $A:=A_{\nu }$ is called the shape operator derived from $\nu$ . The shape operator is metrically equivalent to $h$ , i.e.,

\begin{equation*} g(AX,Y)=h(X,Y) \end{equation*}

holds. The mean curvature function $H$ is defined by $H=(\mathrm {tr}_{g}A)/\dim M$ .

It is known that $M$ is minimal if and only if $H=0$ . The covariant derivative $\nabla h$ of $h$ is defined by:

\begin{equation*} (\nabla h)(Y,Z;\,X)=(\nabla _{X}h)(Y,Z)=Xh(Y,Z)-h(\nabla ^{M}_{X}Y,Z)-h(Y,\nabla ^{M}_XZ). \end{equation*}

The Gauss equation and Codazzi equation are given by

(1) \begin{align} & \tan \left ( R(X,Y)Z \right )=R^{M}(X,Y)Z-A_{h(Y,Z)\nu }X+A_{h(Z,X)\nu }Y, \nonumber \\ & g(R(X,Y)Z,\nu )=(\nabla _{X}h)(Y,Z)-(\nabla _{Y}h)(X,Z), \end{align}

where $R^M$ is the Riemannian curvature of $M$ .

Here, we recall some definitions:

Definition 2.1. A hypersurface $M\subset \widetilde {M}$ is said to be

• • totally geodesic if $h=0$ ,

• • parallel if $\nabla h=0$ ,

• • Codazzi if $\nabla h$ is totally symmetric,

• • totally umbilical if $h=H\,g$ .

Obviously, we have the following inclusion relations.

\begin{equation*} \mbox {totally geodesic}\Longrightarrow \mbox {parallel}\Longrightarrow \mbox {Codazzi} \end{equation*}

A totally umbilical hypersurface $M$ is called an extrinsic sphere if its mean curvature function is a non-zero constant. It should be remarked that $M$ is a Codazzi hypersurface if and only if $R(X,Y)Z$ is tangent to $M$ for all vector fields $X$ , $Y$ and $Z$ on $M$ .

3. The model space $\mathrm {Sol}^4_1$

The model space $\mathrm {Sol}^4_1$ is the Cartesian $4$ -space $\mathbb {R}^4(x,y,z,t)$ equipped with Tricerri metric [Reference Tricerri26]:

\begin{equation*}g=e^{-2t}dx^2+e^{2t}dy^2+(dz-xdy)^2+dt^{\,2}. \end{equation*}

The Riemannian metric $g$ is left invariant with respect to the solvable Lie group structure

\begin{equation*} (x_1,y_1,z_1,t_1) (x_2,y_2,z_2,t_2) =(x_1+e^{t_1}x_2, y_1+e^{-t_1}y_2, z_1+z_2+e^{-t_1}x_1y_2, t_1+t_2). \end{equation*}

Note that the underlying manifold of the model space $\mathrm {Sol}^4_1$ is the solvable Lie group $G_8$ given in [Reference Filipkiewicz18, p. 101].

Here, we explain the reason for our interest in the Tricerri metric. Since $\mathrm {Sol}^4_1$ is $4$ -dimensional, we look for left-invariant Riemannian metric from both a Riemannian and a complex geometric viewpoint.

The moduli space of left-invariant metrics on $\mathrm {Sol}_1^4$ is $5$ -dimensional [Reference Van Thuong30, Theorem 5.1]. Lee and Van Thuong [Reference Lee and Thuong22] proved that a maximal compact subgroup of $\mathrm {Aut}(\mathrm {Sol}^4_1)$ is the dihedral group $\mathrm {D}_4$ of 8 elements [Reference Van Thuong30, Lemma 4.3]. Furthermore, it is unique up to conjugation. Thus, among all left-invariant Riemannian metrics on the underlying solvable Lie group of $\mathrm {Sol}^4_1$ , we first investigate those metrics whose isometry groups contain $\mathrm {D}_4$ (see e.g., [Reference Van Thuong29]). There is a two-parameter family of left-invariant metrics with full isometry group $\mathrm {Sol}^4_1\rtimes \mathrm {D}_4$ (see [Reference Ayad and Fahlaoui1,Reference D’haene12]).

Wall [Reference Wall31] proved that $\mathrm {Sol}^4_1$ admits a compatible complex structure $J$ which will be introduced below. Thus we may look for left-invariant metrics which is compatible with the complex structure $J$ in the $2$ -parameter family mentioned above. By this requirement, we get the Tricerri metric $g$ .

One can take a left-invariant orthonormal frame field with respect to the Tricerri metric $g$ :

\begin{equation*} e_1=e^{t}\frac {\partial }{\partial x}, \quad e_2=e^{-t}\left (\frac {\partial }{\partial y}+x\frac {\partial }{\partial z}\right ), \quad e_3=\frac {\partial }{\partial z}, \quad e_4=\frac {\partial }{\partial t}. \end{equation*}

The Levi-Civita connection $\nabla$ is described as

\begin{equation*} \begin{array}{llll} \nabla _{e_1}e_1=e_4, &\nabla _{e_1}e_2=\frac {1}{2}e_3, & \nabla _{e_1}e_3=-\frac {1}{2}e_2, & \nabla _{e_1}e_4=-e_1, \\[4pt] \nabla _{e_2}e_1=-\frac {1}{2}e_3, & \nabla _{e_2}e_2=-e_4, & \nabla _{e_2}e_3=\frac {1}{2}e_1, & \nabla _{e_2}e_4=e_2, \\[4pt] \nabla _{e_3}e_1=-\frac {1}{2}e_2, &\nabla _{e_3}e_2=\frac {1}{2}e_1, & \nabla _{e_3}e_3=0, & \nabla _{e_3}e_4=0, \\[4pt] \nabla _{e_4}e_1=0, &\nabla _{e_4}e_2=0, & \nabla _{e_4}e_3=0, & \nabla _{e_4}e_4=0. \end{array} \end{equation*}

Note that $e_1$ , $e_2$ , $e_3$ , and $e_4$ are divergence free.

The commutation relations of the $\{e_1,e_2,e_3,e_4\}$ are

\begin{equation*} [e_1,e_2]=e_3, \quad [e_1,e_4]=-e_1, \quad [e_2,e_4]=e_2. \end{equation*}

The non-zero components of Riemannian curvature $R(X,Y)=[\nabla _X,\nabla _Y]-\nabla _{[X,Y]}$ , up to symmetries $R(e_i,e_j)e_k=-R(e_j,e_i)e_k$ and $g(R(e_i,e_j)e_k,e_l)=-g(R(e_i,e_j)e_l,e_k)$ , are identified as follows:

(2) \begin{equation} \begin{array}{lll} R(e_1,e_2)e_2=\frac {1}{4}e_1,& R(e_1,e_3)e_2=\frac {1}{2}e_4,& R(e_1,e_3)e_3=\frac {1}{4}e_1, \\[10pt] R(e_1,e_4)e_1=e_4,& R(e_1,e_4)e_2=\frac {1}{2}e_3,& R(e_2,e_3)e_1=\frac {1}{2}e_4, \\[10pt] R(e_2,e_3)e_3=\frac {1}{4}e_2,& R(e_2,e_4)e_1=\frac {1}{2}e_3,& R(e_2,e_4)e_2=e_4. \end{array} \end{equation}

The sectional curvatures $K_{ij}=K(e_i\wedge e_j)$ of a tangent plane spanned by $e_i$ and $e_j$ are given by

\begin{equation*} K_{12}=K_{13}=K_{23}=\frac {1}{4}, \quad K_{24}=K_{14}=-1, \quad K_{34}=0. \end{equation*}

The scalar curvature is $-\frac {5}{2}$ .

If a left-invariant complex structure $J$ is introduced by

\begin{equation*} Je_1=e_3,\quad Je_2=e_4,\quad Je_3=-e_1,\quad Je_4=-e_2, \end{equation*}

then the Hermitian surface $(\mathrm {Sol}^4_1,g,J)$ is the universal covering of the Inoue surfaces of type $S^{\pm }$ (see [Reference Inoue20]). Sawai [Reference Sawai25] gave a characterization of $\mathrm {Sol}^4_1$ and Inoue surface of type $S^{+}$ among $4$ -dimensional LCK solvmanifolds. Note that $\mathrm {Sol}^4_0$ is the universal covering of Inoue surfaces of type $S^{0}$ .

4. Codazzi hypersurfaces of $\mathrm {Sol}_1^4$

We prove the following non-existence theorem of Codazzi hypersurfaces.

Proof. Let $M$ be a Codazzi hypersurface of $\mathrm {Sol}_1^4$ and assume that its unit normal vector field takes the form $\nu =a e_1 +b e_2 +c e_3 +d e_4$ for some local smooth functions $a, b, c$ and $d$ on $M$ , satisfying $a^2+b^2+c^2+d^2=1$ . Then the following three vector fields form a local orthonormal frame field on $M$ :

(3) \begin{align} T_1=&b e_1 -a e_2 +d e_3 -c e_4, \\ T_2=&c e_1 -d e_2 -a e_3 +b e_4, \nonumber \\ T_3=&d e_1 +c e_2 -b e_3 -a e_4. \nonumber \end{align}

The Codazzi equation (1) for Codazzi hypersurfaces reduces to

(4) \begin{equation} g(R(X,Y)Z, \nu )=0 \end{equation}

for $X, Y, Z \in \mathfrak {X}(M)$ . In particular, taking $X = T_1, Y = T_2$ and $Z = T_3$ , by use of equation (2), we obtain

\begin{equation*} \frac {1}{2}\left [(a-b)^{2}+(c-d)^{2} \right ] \left [(a+b)^{2}+(c+d)^{2}\right ]=0. \end{equation*}

Hence, $b=\pm a$ and $d=\pm c$ . Using these equalities by applying again equation (1) to $X = T_1, Y = T_2$ and $Z = T_2$ , we have

\begin{equation*} \frac {5}{2}d^{2} \left ( b^{2} +d^{2} \right )=0. \end{equation*}

Hence, $c=d=0$ and $|a|=|b|=\frac {1}{\sqrt {2}}$ .

Therefore, the unit normal vector field $\nu$ is chosen as $\nu =\frac {1}{\sqrt {2}}(e_1\pm e_2)$ and the tangent bundle $TM$ is generated by the distribution spanned by $E_1=\frac {1}{\sqrt {2}}(e_1\mp e_2)$ , $E_2=e_3$ and $E_3=e_4$ .

However, this distribution is not integrable, indeed we have

\begin{equation*} [E_1,E_2]=0,\quad [E_1,E_3]=-\nu, \quad [E_2,E_3]=0. \end{equation*}

We now present the following corollary.

5. Totally umbilical hypersurfaces in $\mathrm {Sol}_1^4$

The existence of proper totally umbilical hypersurface is a strong restriction for Riemannian symmetric spaces. Chen [Reference Chen2] proved that if a Riemannian symmetric space of dimension $\geq 4$ admits non-totally geodesic totally umbilical hypersurfaces then it is conformally flat. In this section, we prove the non-existence of totally umbilical hypersurfaces in $\mathrm {Sol}_1^4$ .

Let $M$ be a totally umbilical hypersurface in $\mathrm {Sol}_1^4$ . We define its unit normal vector field $\nu$ and tangent vector fields $T_1, T_2$ and $T_3$ as in the previous section (see (3)). The Codazzi equation (1) for totally umbilical hypersurfaces is

(5) \begin{equation} g(R(X,Y)Z, \nu )=g(Y,Z)X(H)-g(Z,X)Y(H), \qquad X,Y,Z \in \mathfrak {X}(M). \end{equation}

In particular, substituting $X = T_1, Y = T_2$ and $Z = T_3$ into equation (5), we have again

\begin{equation*} \frac {1}{2}\left [(a-b)^{2}+(c-d)^{2} \right ] \left [(a+b)^{2}+(c+d)^{2}\right ]=0. \end{equation*}

Hence, $b=\pm a$ and $d=\pm c$ . Using these equalities by applying again equation (5) to $X = T_1, Y = T_2, Z = T_1$ , (and also $X = T_2, Y = T_3, Z = T_3$ ), we have

\begin{equation*} \frac {5}{2}ac \left (a^{2} +c^{2} \right )=-T_2 (H) \end{equation*}

Applying (5) to $X = T_1, Y = T_2, Z = T_2$ , (and $X = T_3, Y = T_1, Z = T_3$ , too) we obtain

\begin{equation*} \frac {5}{2}c^{2} \left ( a^{2} +c^{2} \right )=T_1 (H) \end{equation*}

If we apply equation (5) to $X = T_3, Y = T_1, Z = T_1$ , (and also $X = T_2, Y = T_3, Z = T_2$ ), we obtain the following:

\begin{equation*} \frac {1}{2}ac \left (a^{2} +c^{2} \right )=T_3 (H). \end{equation*}

Next, we check the integrability conditions. First, we express the orthonormal frame field $\{e_1,e_2,e_3,e_4\}$ of the ambient space by local orthonormal frame field $\{T_1,T_2,T_3\}$ along $M$ and the unit normal vector field $\nu$ , by using equation (3), $b=\pm a$ and $d=\pm c$ . It suffices to investigate the case $b=a$ and $d=c$ . We have

(6) \begin{align} e_1=& a \nu +a T_1 + c T_2 +c T_3, \\ e_2=& a \nu -a T_1 - c T_2 +c T_3, \nonumber \\ e_3=& c \nu +c T_1 - a T_2 -a T_3, \nonumber \\ e_4=& c \nu -c T_1 + a T_2 -a T_3. \nonumber \end{align}

Computing $[T_1,T_2]$ for $b=a$ and $d=c$ , we obtain

\begin{align*} [T_1,T_2]=&\left (T_1 (c)-T_2(a) -(a^{2}+c^{2}) \right )e_1 - \left (T_1 (c)-T_2(a) +(a^{2}+c^{2}) \right )e_2 + \\ &- \left (T_1 (a)+T_2(c) \right )e_3 + \left (T_1 (a)+T_2(c) \right )e_4. \end{align*}

Substituting equation (6) into this expression, using $2(a^{2}+c^{2})=1$ , we obtain that the normal component of $[T_1,T_2]$ is $-a \nu$ . Hence, it follows $a=0$ .

Analogously, computing $[T_1,T_3]$ for $b=a$ and $d=c$ , we obtain

\begin{align*} [T_1,T_3]=&\quad \left (T_1 (c)-T_3(a)+a^{2} -c^{2} \right )e_1 + \left (T_1 (c)+T_3(a) +a^{2} +c^{2} \right )e_2 \\ &+ \left (-T_1 (a)-T_3(c)+2ac \right )e_3 + \left (-T_1 (a)+T_3(c) \right )e_4. \end{align*}

Substituting equation (6) into this expression, we find that the normal component of $[T_1,T_3]$ is $\left (a +2aT_1(c)-2cT_1(a)\right ) \nu$ . Since it is known that $[T_1, T_3]$ has no normal component, it follows that $a = 0$ .

Finally, we have

\begin{align*} [T_2,T_3]=&\left (2ac+T_2 (c)-T_3(c) \right )e_1 + \left (T_2 (c)+T_3(c) \right )e_2 + \\ &+ \left (2c^{2}-T_2 (a)+T_3(a) \right )e_3 + \left (-T_2 (a)-T_3(a) \right )e_4, \end{align*}

and the normal component of $[T_2,T_3]$ is $\left (c +2aT_2(c)-2cT_2(a)\right ) \nu$ . Hence, it follows $c=0$ . This contradicts to $a^2+c^2=1/2$ . Therefore, the distribution spanned by $\{T_1,T_2, T_3\}$ is not integrable and we proved the following theorem.

Our results give a complete answer to the problem posed in [Reference Inoguchi19, p. 126, Problem 1].

As we mentioned, the moduli space of left-invariant metrics on $\mathrm {Sol}^4_1$ is $5$ -dimensional. In particular, there exists a $2$ -parameter family of left-invariant metrics with full isometry group $\mathrm {Sol}^4_1\rtimes \mathrm {D}_4$ . There are three possible cases for the full isometry group: $\mathrm {Sol}^4_1\rtimes \mathrm {D}_4$ , $\mathrm {Sol}^4_1\rtimes \mathbb {Z}_2$ or $\mathrm {Sol}^4_1$ (see [Reference Ayad and Fahlaoui1]).

As a first step in the study of hypersurface geometry of $\mathrm {Sol}^4_1$ , we focus our attention on the Tricerri metric in this article. In a forthcoming article, we will investigate Codazzi and totally umbilical hypersurfaces in $\mathrm {Sol}^4_1$ with respect to general left-invariant metric.

We conclude this article with the following problem: Determine the left-invariant metrics on $\mathrm {Sol}^4_1$ that admit totally geodesic hypersurfaces.