On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group

Abstract

In total, geodesic surfaces and their generalizations, namely totally umbilical and parallel surfaces, are well-known topics in Submanifold Theory and have been intensively studied in three-dimensional ambient spaces, both Riemannian and Lorentzian. In this paper, we prove the non-existence of parallel and totally umbilical (in particular, totally geodesic) surfaces for three-dimensional Lorentzian Lie groups, which admit a four-dimensional isometry group, but are neither of Bianchi–Cartan–Vranceanu-type nor homogeneous plane waves. Consequently, the results of the present paper complete the investigation of these fundamental types of surfaces in all homogeneous Lorentzian manifolds, whose isometry group is four-dimensional. As a byproduct, we describe a large class of flat surfaces of constant mean curvature in these ambient spaces and exhibit a family of examples.

1. Introduction

The study of three-dimensional homogeneous manifolds is a very active area of research in differential geometry, having drawn the attention of many scholars, due to the richness of their geometric properties. A nondegenerate metric in a three-dimensional pseudo-Riemannian manifold must be (up to reversing its sign [1]) either Riemannian or Lorentzian. Notably, Lorentzian settings tend to exhibit a broader spectrum of geometric behaviors, many of which have no analogue in the Riemannian context.

A key aspect in the analysis of these spaces lies in understanding their isometry groups. For a three-dimensional pseudo-Riemannian manifold $(M,g)$, the group of isometries $\mathrm{Iso}(M,g)$ has a dimension of at most six. The case where $\dim \left(\mathrm{Iso}\right(M,g\left)\right)=6$ characterizes manifolds of constant sectional curvature. It is also known that there are no three-dimensional pseudo-Riemannian manifolds whose isometry group has a dimension of five.

This naturally motivates the study of pseudo-Riemannian three-manifolds $(M,g)$ admitting a four-dimensional isometry group. In such cases, the action of $\mathrm{Iso}(M,g)$ is transitive, which implies that $(M,g)$ is a homogeneous manifold.

The Riemannian setting has been completely studied and is now well understood. Three-dimensional Riemannian manifolds with a four-dimensional isometry group are referred to as Bianchi–Cartan–Vranceanu spaces (often abbreviated as BCV spaces), named after the three mathematicians who made pioneering works on these spaces from distinct points of view (see [2,3,4,5]).

As it is often the case, the Lorentzian case offers more possibilities than its Riemannian counterpart. Indeed, some different types of homogeneous Lorentzian three-manifolds possess a four-dimensional isometry group. This becomes evident when one compares the classification of non-isometric homogeneous structures in the Riemannian setting [6] with the one in the Lorentzian context [7]. Specifically, as shown in [7], three-dimensional Lorentzian Lie groups whose isometry group is four-dimensional form three mutually exclusive classes, which exist for some distinct forms of the self-adjoint structure operator L associated with their Lie algebra:

(I)

Lorentzian Bianchi–Cartan–Vranceanu spaces, for which L is diagonal, show a structure similar to Riemannian BCV spaces (although the Lorentzian class presents more cases than its Riemannian analogue) and exhibit several interesting geometric properties. In particular, they are naturally reductive spaces and are defined by submersions over a pseudo-Riemannian surface of constant curvature [8].

(II)

An exceptional example $\mathcal{E}$, for which the minimal polynomial of L has a non-zero double root. At the Lie algebra level, this Lorentzian Lie group is described by

$$[u_1,u_2]=\mu u_3,[u_2,u_3]=\mu u_2,[u_3,u_1]=\mu u_1+\epsilon u_2,\mu \ne 0,\epsilon =\pm 1,$$

(1)

where $\{u_1,u_2,u_3\}$ is a pseudo-orthonormal basis with $u_3$ being spacelike, i.e., such that the Lorentzian inner product $⟨,⟩$ is completely determined by the non-vanishing components

$$⟨u_1,u_2⟩=⟨u_3,u_3⟩=1.$$

(2)

Equations (1) and (2) describe a one-parameter family of left-invariant Lorentzian metrics on $G=\widetilde{SL}(2,\mathbb{R})$ (we may refer either to [7] or to Section 3 for more details). These examples are still naturally reductive but do not have a structure similar to Lorentzian BCV spaces.

(III)

Some homogeneous plane waves, for which the minimal polynomial of L has 0 as a triple root. Homogeneous plane waves are a well-known topic, whose study which dates back to the work by [9] in the framework of theoretical physics. These examples are not naturally reductive.

We refer to the above case (Case II) as “exceptional” with respect to Case (I), which admits a Riemannian analogue, and Case (III), which is well-known in the literature.

In this framework, the investigation of immersed surfaces plays a fundamental role. Exploring the geometric features of surfaces in homogeneous spaces provides deeper insights into the ambient manifold’s global structure and enables the classification of surfaces through intrinsic and extrinsic criteria, such as total umbilicity, the parallelism of the second fundamental form, and the constancy of mean curvature. The investigation of such surfaces often reveals essential features of the ambient homogeneous space, serving as a powerful geometric tool to distinguish between different types of ambient spaces. This is true both for the cases where examples occur and classifications are achieved, and for the ones where non-existence results of submanifolds with the required properties are proved (see, for example, [10,11]). An exhaustive list of references concerning the geometry of surfaces in homogeneous Riemannian three-manifolds will dive into an investigation beyond the possibilities and purposes of the actual work. Besides the ones we already cited, a few works in this context are given by [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29] and the references therein.

In Lorentzian settings, several authors have studied some remarkable families of surfaces in some ambient spaces that have a correspondence with the Riemannian cases, such as Lorentzian Berger spheres [30] and the Heisenberg group endowed with some left-invariant Lorentzian metrics [31,32,33]. In recent works ([8,34]), the authors of this paper described parallel, totally geodesic and totally umbilical surfaces of the non-exceptional cases—Cases (I) and (III)—of three-dimensional Lorentzian manifold admitting a group of isometry with a dimension of four, providing some examples of minimal and CMC surfaces. Together with M. Castrillon-Lopez [31], we investigated totally umbilical and minimal surfaces of the Lorentzian Heisenberg groups, proving the existence of a useful relation between minimality and total umbilicity in the Lorentzian context, which involves the conformality of the Gauss map of the surface. Moreover, the complete characterization of totally umbilical surfaces of Lorentzian reducible spaces was provided in [35].

Taking into account the results of [8,34,35], in order to complete the investigation of remarkable surfaces in Lorentzian homogeneous three-manifolds with a four-dimensional isometry group, one needs to consider these surfaces in the exceptional example $\mathcal{E}$. This research is the aim of this paper.

The paper is organized in the following way. In Section 2, we report some basic information concerning the geometric properties of $\mathcal{E}$ and the geometry of surfaces in a Lorentzian ambient space. In Section 3, we prove the non-existence of parallel (in particular, totally geodesic) surfaces of $\mathcal{E}$. On the other hand, the study of the case of parallel surfaces is a special case of surfaces with a Codazzi second fundamental form. We investigate the broader class of Codazzi surfaces, which turn out to be flat, and CMC surfaces, which exhibit a family that can be used as an example. In Section 4, we prove the non-existence of totally umbilical surfaces in $\mathcal{E}$.

2. Preliminaries

2.1. The Geometry of the Exceptional Example

For any choice of the real parameter $\mu \ne 0$, the unimodular Lie group $G=\widetilde{SL}(2,\mathbb{R})$ with the left-invariant Lorentzian metric determined by (1) and (2) is isometric to ${\mathbb{R}}^3$, equipped with the metric

$$g={\displaystyle \frac{\epsilon }{\mu }}(e^{-2\mu x_3}-1)d{x}_1^2+2d{x}_{1}d{x}_2+2\mu x_{2}d{x}_{1}d{x}_3+d{x}_3^2,\epsilon =\pm 1.$$

(3)

In order to simplify the notation, throughout this paper, we shall use $\mathcal{E}$ to denote the exceptional example with four-dimensional isometry group, i.e., (an open subset of) ${\mathbb{R}}^3$ equipped with the Lorentzian metric (3).

We shall use the notation ${\partial }_i=\partial /\partial x_i$, for all indices i. As ${\partial }_i$, $i=1,2,$ and 3 are global coordinate vector fields on $\mathcal{E}$, the well-known Koszul formula simplifies as

$$2g({\nabla }_{{\partial }_i}{\partial }_j,{\partial }_k)={\partial }_i\left(g({\partial }_j,{\partial }_k)\right)+{\partial }_j\left(g({\partial }_i,{\partial }_k)\right)-{\partial }_k\left(g({\partial }_j,{\partial }_i)\right).$$

(4)

and allows us to determine ${\nabla }_{{\partial }_i}{\partial }_j$ for all indices $i,j$. For example, starting from (3) and setting $i=2$ and $j=3$ in (4), we get

$$\left\{\begin{array}{c}2g({\nabla }_{{\partial }_2}{\partial }_3,{\partial }_1)=\mu ,\hfill \\ 2g({\nabla }_{{\partial }_2}{\partial }_3,{\partial }_2)=0,\hfill \\ 2g({\nabla }_{{\partial }_2}{\partial }_3,{\partial }_3)=0\hfill \end{array}\right.$$

whence, ${\nabla }_{{\partial }_2}{\partial }_3={\nabla }_{{\partial }_3}{\partial }_2={\displaystyle \frac{1}{2}}\mu {\partial }_2$. Similar calculations allow us to describe the Levi-Civita connection ∇ of g, which is then explicitly determined by the following possibly non-vanishing components:

$$\begin{array}{c}{\nabla }_{{\partial }_1}{\partial }_1=-\epsilon \mu x_2{e}^{-2\mu x_3}{\partial }_2+\epsilon e^{-2\mu x_3}{\partial }_3,\hfill \\ {\nabla }_{{\partial }_1}{\partial }_2={\nabla }_{{\partial }_2}{\partial }_1=-{\displaystyle \frac{1}{2}}{\mu }^2{x}_2{\partial }_2+{\displaystyle \frac{1}{2}}\mu {\partial }_3,\hfill \\ {\nabla }_{{\partial }_1}{\partial }_3={\nabla }_{{\partial }_3}{\partial }_1=-{\displaystyle \frac{1}{2}}\mu {\partial }_1-{\displaystyle \frac{1}{2}}(\epsilon e^{-2\mu x_3}+\epsilon +{\mu }^3{x}_2^2){\partial }_2+{\displaystyle \frac{1}{2}}{\mu }^2{x}_2{\partial }_3,\hfill \\ {\nabla }_{{\partial }_2}{\partial }_3={\nabla }_{{\partial }_3}{\partial }_2={\displaystyle \frac{1}{2}}\mu {\partial }_2.\hfill \end{array}$$

(5)

Next, we consider the curvature tensor R, taken with the sign convention $R(X,Y)={\nabla }_{[X,Y]}-[{\nabla }_X,{\nabla }_Y]$. A direct calculation yields that, for example,

$$R({\partial }_2,{\partial }_3){\partial }_3={\nabla }_{{\partial }_2}{\nabla }_{{\partial }_3}{\partial }_3-{\nabla }_{{\partial }_3}{\nabla }_{{\partial }_2}{\partial }_3=-{\displaystyle \frac{1}{2}}\mu {\nabla }_{{\partial }_3}{\partial }_2={\displaystyle \frac{1}{4}}{\mu }^2{\partial }_2.$$

With similar calculations, we find that the curvature tensor R is completely determined, up to symmetries, by the following possibly non-vanishing components:

$$\begin{array}{c}R({\partial }_1,{\partial }_2){\partial }_1={\displaystyle \frac{1}{4}}{\mu }^2{\partial }_1-{\displaystyle \frac{1}{4}}\epsilon \mu (e^{-2\mu x_3}-1){\partial }_2,\hfill \\ R({\partial }_1,{\partial }_2){\partial }_2=-{\displaystyle \frac{1}{4}}{\mu }^2{\partial }_2,\hfill \\ R({\partial }_1,{\partial }_2){\partial }_3=-{\displaystyle \frac{1}{4}}{\mu }^3{x}_2{\partial }_2,\hfill \\ R({\partial }_1,{\partial }_3){\partial }_1={\displaystyle \frac{1}{4}}{\mu }^3{x}_2{\partial }_1+\epsilon {\mu }^2{x}_2{e}^{-2\mu x_3}{\partial }_2-{\displaystyle \frac{1}{4}}\epsilon \mu (5e^{-2\mu x_3}-1){\partial }_3,\hfill \\ R({\partial }_1,{\partial }_3){\partial }_2=-{\displaystyle \frac{1}{4}}{\mu }^2{\partial }_3,\hfill \\ R({\partial }_1,{\partial }_3){\partial }_1={\displaystyle \frac{1}{4}}{\mu }^2{\partial }_1+\epsilon \mu e^{-2\mu x_3}{\partial }_2-{\displaystyle \frac{1}{4}}{\mu }^3{x}_2{\partial }_3,\hfill \\ R({\partial }_2,{\partial }_3){\partial }_1={\displaystyle \frac{1}{4}}{\mu }^3{x}_2{\partial }_2-{\displaystyle \frac{1}{4}}{\mu }^2{\partial }_3,\hfill \\ R({\partial }_2,{\partial }_3){\partial }_3={\displaystyle \frac{1}{4}}{\mu }^2{\partial }_2.\hfill \end{array}$$

(6)

In terms of the coordinate vector fields, we have the following explicit expression for the pseudo-orthonormal basis $\left\{u_i\right\}$:

$$\left\{\begin{array}{c}{u}_1=e^{\mu x_3}{\partial }_1+{\displaystyle \frac{1}{2\mu }}\left(({\mu }^3{x}_2^2+\epsilon )e^{\mu x_3}-\epsilon e^{-\mu x_3})\right){\partial }_2-\mu x_2{e}^{\mu x_3}{\partial }_3,\hfill \\ u_2=e^{-\mu x_3}{\partial }_2,\hfill \\ u_3={\partial }_3.\hfill \end{array}\right.$$

(7)

Starting from (5) and (7), we can also determine the Levi-Civita connection as follows:

$$\begin{array}{ccc}{\nabla }_{u_1}{u}_1=\epsilon u_3,\hfill & {\nabla }_{u_2}{u}_1=-{\displaystyle \frac{\mu }{2}}{u}_3,\hfill & {\nabla }_{u_3}{u}_1={\displaystyle \frac{\mu }{2}}{u}_1,\hfill \\ {\nabla }_{u_1}{u}_2={\displaystyle \frac{\mu }{2}}{u}_3,\hfill & {\nabla }_{u_2}{u}_2=0,\hfill & {\nabla }_{u_3}{u}_2=-{\displaystyle \frac{\mu }{2}}{u}_2,\hfill \\ {\nabla }_{u_1}{u}_3=-{\displaystyle \frac{\mu }{2}}{u}_1-\epsilon u_2,\hfill & {\nabla }_{u_2}{u}_3={\displaystyle \frac{\mu }{2}}{u}_2,\hfill & {\nabla }_{u_3}{u}_3=0.\hfill \end{array}$$

(8)

Moreover, we observe that the Killing vector field $u_2$ satisfies

$${\nabla }_X{u}_2=-{\displaystyle \frac{\mu }{2}}(X\wedge u_2),$$

(9)

for all $X\in T\mathcal{E}$, where the wedge product ∧ is completely determined by

$$u_1\wedge u_2=-u_3,u_2\wedge u_3=-u_2,u_1\wedge u_3=u_1$$

(10)

(see [36]).

From (6) and (7), we deduce that the curvature tensor is completely determined by

$$\begin{array}{ccc}R(u_1,u_2)u_1={\displaystyle \frac{{\mu }^2}{4}}{u}_1,\hfill & R(u_1,u_2)u_2=-{\displaystyle \frac{{\mu }^2}{4}}{u}_2,\hfill & \\ R(u_1,u_3)u_1=-\epsilon \mu u_3,\hfill & R(u_1,u_3)u_2=-{\displaystyle \frac{{\mu }^2}{4}}{u}_3,\hfill & R(u_1,u_3)u_3={\displaystyle \frac{{\mu }^2}{4}}{u}_1+\epsilon \mu u_2,\hfill \\ R(u_2,u_3)u_1=-{\displaystyle \frac{{\mu }^2}{4}}{u}_3,\hfill & R(u_2,u_3)u_3={\displaystyle \frac{{\mu }^2}{4}}{u}_2.\hfill \end{array}$$

(11)

Next, we can prove the following result.

Proposition 1. 

The curvature tensor of $\mathcal{E}$ is given by

$$R(X,Y)Z=-{\displaystyle \frac{{\mu }^2}{4}}\left(g(X,Z)Y-g(Y,Z)X\right)+\epsilon \mu g\left(X\wedge Y,u_2\right)Z\wedge u_2$$

for all tangent vector fields $X,Y,Z$.

Proof. 

Let $X,Y,Z$ be three arbitrary vector fields on $\mathcal{E}$. We write them as

$$\begin{array}{c}\hfill X=\overline{X}+x{u}_3,Y=\overline{Y}+y{u}_3,Z=\overline{Z}+z{u}_3,\end{array}$$

with $\overline{X},\overline{Y},\overline{Z}$ orthogonal to $u_3$ and $x=g(X,u_3)$, and so on. Observe that from (11), it follows that all terms of $g\left(R\right(X,Y)Z,W)$, where $u_3$ occurs either one, three or four times, necessarily vanish. Therefore, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill g\left(R\right(X,Y)Z,W)& =g(R(\overline{X},\overline{Y})\overline{Z},\overline{W})\hfill \\ \hfill & +xzg(R(u_3,\overline{Y})u_3,\overline{W})+yzg(R(\overline{X},u_3)u_3,\overline{W})\hfill \\ \hfill & +ywg(R(\overline{X},u_3)\overline{Z},u_3)+xwg(R(u_3,\overline{Y})\overline{Z},u_3).\hfill \end{array}\end{array}$$

From the decompositions of $\overline{X}$, $\overline{Y}$, $\overline{Z}$ and $\overline{W}$, it follows that

$$\begin{array}{c}\hfill g(R(\overline{X},\overline{Y})\overline{Z},\overline{W})=-{\displaystyle \frac{{\mu }^2}{4}}(g(\overline{X},\overline{Z})g(\overline{Y},\overline{W})-g(\overline{X},\overline{W})g(\overline{Y},\overline{Z})).\end{array}$$

Moreover, we have

$$\begin{array}{c}\hfill R(\overline{X},u_3)u_3={\displaystyle \frac{{\mu }^2}{4}}\overline{X}+\epsilon \mu g(\overline{X},u_2)u_2,R(u_3,\overline{Y})u_3=-{\displaystyle \frac{{\mu }^2}{4}}\overline{Y}-\epsilon \mu g(\overline{Y},u_2)u_2.\end{array}$$

Hence, we conclude that

$$\begin{array}{cc}\hfill g\left(R\right(X,Y)Z,W)=& -{\displaystyle \frac{{\mu }^2}{4}}(g(\overline{X},\overline{Z})g(\overline{Y},\overline{W})-g(\overline{X},\overline{W})g(\overline{Y},\overline{Z}))\hfill \\ \hfill & -{\displaystyle \frac{{\mu }^2}{4}}\left(xzg(\overline{Y},\overline{W})-yzg(\overline{X},\overline{W})\right)+\epsilon \mu z\left(yg(\overline{X},u_2)-xg(\overline{Y},u_2)\right)g(u_2,\overline{W})\hfill \\ \hfill & -{\displaystyle \frac{{\mu }^2}{4}}\left(ywg(\overline{X},\overline{Z})-xwg(\overline{Y},\overline{Z})\right)+\epsilon \mu w\left(xg(\overline{Y},u_2)-yg(\overline{X},u_2)\right)g(u_2,\overline{Z})\hfill \\ \hfill =& -{\displaystyle \frac{{\mu }^2}{4}}(g(X,Z)g(Y,W)-g(X,W)g(Y,Z))\hfill \\ \hfill & +\epsilon \mu \left(yg(X,u_2)-xg(\overline{Y},u_2)\right)\left(g(u_3,Z)g(u_2,W)-g(u_2,Z)g(u_3,W)\right)\hfill \\ \hfill =& g\left(-{\displaystyle \frac{{\mu }^2}{4}}\left(g(X,Z)Y-g(Y,Z)X\right)+\epsilon \mu g\left(X\wedge Y,u_2\right)Z\wedge u_2,W\right),\hfill \end{array}$$

which ends the proof, as W is an arbitrary vector field. □

2.2. Geometry of Surfaces in a Three-Dimensional Lorentzian Ambient Space

Let $F:\mathsf{\Sigma }\to M$ be an isometric immersion, where $(\mathsf{\Sigma },g^{\mathsf{\Sigma }})$ denotes an oriented pseudo-Riemannian surface of a three-dimensional pseudo-Riemannian manifold $(M,g)$, and the metric $g^{\mathsf{\Sigma }}$ on $\mathsf{\Sigma }$ is the pullback of g via the immersion F.

The normal unit vector field N on $\mathsf{\Sigma }$ satisfies $g(N,N)=\delta$, where $\delta \in \{-1,1\}$. If $\delta =1$, then N is spacelike, implying that $\mathsf{\Sigma }$ is a timelike surface and the immersion is Lorentzian. On the other hand, if $\delta =-1$, then N is timelike and, therefore, $\mathsf{\Sigma }$ is spacelike and the immersion is Riemannian.

Let ${\nabla }^{\mathsf{\Sigma }}$ and ∇ denote the Levi-Civita connections of $g^{\mathsf{\Sigma }}$ and g, respectively. Given any vector fields X and Y tangent to $\mathsf{\Sigma }$, these connections are related via the classical Gauss equation:

$${\nabla }_{X}Y={\nabla }_X^{\mathsf{\Sigma }}Y+h(X,Y)N,$$

where h is a symmetric bilinear form on the tangent bundle of $\mathsf{\Sigma }$, known as the second fundamental form associated with the immersion F.

Next, the formula $SX=-{\nabla }_{X}N$ defines the shape operator S of $\mathsf{\Sigma }$ for every tangent vector X. The shape operator S is related to the second fundamental form h by

$$g(SX,Y)=\delta h(X,Y),$$

for all X and Y tangent to $\mathsf{\Sigma }$.

The behavior of h (equivalently, of S) at various points provides insights into the geometry of the submanifold. Specifically, a point $p\in \mathsf{\Sigma }$ is referred to as totally umbilical if there exists some constant $\lambda \left(p\right)$, such that

$$h(X_p,Y_p)=\lambda \left(p\right)g_p^{\mathsf{\Sigma }}(X_p,Y_p)$$

for all tangent vectors $X_p,Y_p\in T_p\mathsf{\Sigma }$. When this condition holds throughout the entire surface, we say that $\mathsf{\Sigma }$ is totally umbilical. This means that around each point, there exists an open set $U\subseteq \mathsf{\Sigma }$ and a smooth function $\lambda :U\to \mathbb{R}$, for which

$$h(X,Y)=\lambda g^{\mathsf{\Sigma }}(X,Y)$$

for all vector fields $X,Y$ defined on U. This geometric condition is equivalent to requiring that the shape operator S is a multiple of the identity (namely $S=\lambda \mathrm{Id}$) at each point. A special case arises when $\lambda =0$. In this case, $h\equiv 0$, meaning that $\mathsf{\Sigma }$ is totally geodesic.

Another geometric property connected to some condition on the second fundamental form of the immersion is given by parallel surfaces. A surface is said to be parallel if its second fundamental form is covariantly constant, i.e.,

$$\nabla h=0.$$

A weaker version of this requirement is the semi-parallel condition. Let $R^{\mathsf{\Sigma }}$ and R be the curvature tensors of $\mathsf{\Sigma }$ and M, respectively. The immersion is called semi-parallel if it satisfies the following:

$$R^{\mathsf{\Sigma }}·h=0,$$

that is, more explicitly,

$$(R^{\mathsf{\Sigma }}·h)(X,Y,Z,W)=-h(R^{\mathsf{\Sigma }}(X,Y)Z,W)-h(Z,R^{\mathsf{\Sigma }}(X,Y)W)=0,$$

for all vector fields $X,Y,Z,W$ tangent to $\mathsf{\Sigma }$.

Any parallel surface is also semi-parallel. Moreover, in the case where the ambient manifold is three-dimensional, the first author and Van der Veken in [37] proved the following relation between semi-parallel and totally umbilical surfaces:

Lemma 1

([37]). A surface Σ in a three-dimensional Lorentzian manifold $(M,g)$ is semi-parallel if and only if it is either flat or totally umbilical.

Further geometric constraints are encoded in the Gauss and Codazzi equations, which relate the intrinsic and extrinsic geometry of $\mathsf{\Sigma }$:

$$\begin{array}{cc}\hfill g\left(R\right(X,Y)Z,W)& =g(R^{\mathsf{\Sigma }}(X,Y)Z,W)+\delta \left(h(X,Z)h(Y,W)-h(X,W)h(Y,Z)\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill g\left(R\right(X,Y)Z,N)& =\delta \left((\nabla h)(X,Y,Z)-(\nabla h)(Y,X,Z)\right),\hfill \end{array}$$

(13)

where $X,Y,Z,W$ are vector fields tangent to $\mathsf{\Sigma }$. The Codazzi condition for the second fundamental form refers to the total symmetry of the covariant derivative $\nabla h$. From Equation (13), one sees that this symmetry condition holds for parallel hyper-surfaces. Moreover, it is equivalent to the following condition:

$$g\left(R\right(X,Y)N,Z)=0$$

(14)

for all vector fields $X,Y,Z$ tangent to $\mathsf{\Sigma }$. Thus, a surface satisfying Equation (14) is said to have a Codazzi second fundamental form (or, briefly, a Codazzi surface).

Among submanifolds, minimal and constant mean curvature (CMC) surfaces have a special interest. The mean curvature H of a surface $\mathsf{\Sigma }$ in a three-dimensional ambient space $\tilde{M}$ is given by

$$H={\displaystyle \frac{1}{2}}{\mathrm{tr}}_{g^{\mathsf{\Sigma }}}h={\displaystyle \frac{1}{2}}\sum {\left(g^{\mathsf{\Sigma }}\right)}^{ij}{h}_{ij},$$

where ${\left(g^{\mathsf{\Sigma }}\right)}^{ij}$ are the coefficients of the inverse metric ${\left(g^{\mathsf{\Sigma }}\right)}^{-1}$, expressed in a local frame. The surface is said to be minimal if $H=0$. (In the context of Lorentzian geometry, for timelike surfaces, the vanishing of the mean curvature is often referred to as maximality). If the mean curvature is constant on $\mathsf{\Sigma }$, the surface is called CMC. Clearly, totally geodesic surfaces are minimal, and minimal surfaces are necessarily CMC.

3. Codazzi Surfaces in $\mathcal{E}$

We shall now investigate surfaces $\mathsf{\Sigma }$ of $\mathcal{E}$ having a Codazzi second fundamental form. As we already mentioned in the Introduction Section, $\mathcal{E}$ is the class of three-dimensional Lorentzian Lie groups with a four-dimensional isometry group, such that the minimal polynomial of the structure operator L of its Lie algebra has a non-vanishing double root.

In fact, using (1) and (10), we easily obtain that L, completely determined by $L(u_i\wedge u_j)=[u_i,u_j]$, has eigenvalues ${\lambda }_1={\lambda }_2=-\mu$ (with a one-dimensional associated eigenspace) and ${\lambda }_3=\mu$.

The general form of the Lie algebra of a three-dimensional Lorentzian Lie group, with the structure operator L admitting a double root, is the following: its Lie algebra admits an orthonormal basis $\{e_1,e_2,e_3\}$, with $e_3$ being timelike, such that

$$\begin{array}{ccc}\hfill & & \left[e_1,e_2\right]=-e_2+(2\eta -\beta )e_3,\eta =\pm 1,\hfill \\ \hfill \left({\mathfrak{g}}_4\right):& & \left[e_1,e_3\right]=-\beta e_2+e_3,\hfill \\ & & \left[e_2,e_3\right]=\alpha e_1.\hfill \end{array}$$

(15)

The eigenvalues of L are given by ${\lambda }_1={\lambda }_2=\beta -\eta$ (with a one-dimensional associated eigenspace) and ${\lambda }_3=\alpha$. Therefore, $\mathcal{E}$ corresponds to the special case of (15), where $\alpha =\mu \ne 0$ and $\beta -\eta =-\mu \ne 0$, yielding a family of left-invariant metrics on $\widetilde{SL}(2,\mathbb{R})$ [36].

Parallel surfaces of all three-dimensional Lorentzian Lie groups, whose structure operator L has a double root, were already classified in the following result of [37].

Theorem 1

([37]). Let M be a parallel surface in a non-symmetric three-dimensional Lorentzian Lie group with Lie algebra (15). Then, one of the following statements holds:

 (a) 

M is an integral surface of the distribution spanned by $\{e_2,e_3\}$. This case only occurs if $\alpha =0$. M is parallel, flat and minimal, but not totally geodesic.

 (b) 

M is an integral surface of the distribution spanned by $\{e_1,c{e}_2+b{e}_3\}$, where b and c are constants satisfying $b^2-c^2=\delta$ and $\beta b^2+2bc+(\beta -2\eta )c^2=0$. M is totally geodesic and has constant Gaussian curvature $K=-\delta (\beta -\eta )$.

We now consider a pseudo-Riemannian-oriented surface $\mathsf{\Sigma }$ immersed into $\mathcal{E}$, whose normal vector field N satisfies $g(N,N)=\delta =\pm 1$. Using the notation we introduced in the previous section, the following result easily follows from Proposition 1.

Corollary 1. 

For any $X,Y$ tangent to Σ, we have the following:

$$R(X,Y)N=\epsilon \mu g(X\wedge Y,u_2)(N\wedge u_2).$$

(16)

We now prove the following.

Theorem 2.

Let Σ denote a surface in the exceptional example $\mathcal{E}$ with a Codazzi second fundamental form. Then,

 (I) 

Every point of Σ admits an open neighborhood $U\subseteq \mathsf{\Sigma }$, where $N=\varphi u_2+u_3$, for some smooth function $\varphi :U\to \mathbb{R}$. In particular, Σ is timelike.

 (II) 

Σ is flat.

 (III) 

Σ is CMC but never minimal.

 (IV) 

Σ is never parallel (in particular, totally geodesic).

Proof. 

Starting from (7) and (2), we can easily deduce an orthonormal basis for $\mathcal{E}$, namely

$$\begin{array}{c}\hfill E_1={\displaystyle \frac{u_1+u_2}{\sqrt{2}}},E_2={\displaystyle \frac{u_1-u_2}{\sqrt{2}}},E_3=u_3,\end{array}$$

(17)

where $g(E_1,E_1)=g(E_3,E_3)=-g(E_2,E_2)=1$.

In a neighborhood U of some point $p\in \mathsf{\Sigma }$, we consider $N=A{E}_1+B{E}_2+C{E}_3$, for some functions $A,B,C:U\to \mathbb{R}$ such that $g(N,N)=A^2-B^2+C^2=\delta =\pm 1\ne 0$. Then, the following vector fields are tangent to the surface:

$$X_1=B{E}_1+A{E}_2,X_2=C{E}_1-A{E}_3,X_3=C{E}_2+B{E}_3.$$

If h is Codazzi, then Equation (13) yields that $g(R(X_i,X_j)N,X_k)=0$ for all indices $i,j,k\in \{1,2,3\}$. Hence, using the formula we introduced in Proposition 1, we have

$$\begin{array}{cc}0\hfill & =g(R(X_1,X_2)N,X_k)=\epsilon \mu Ag(N,u_2)g(N\wedge u_2,X_k)=\epsilon \mu {\displaystyle \frac{A}{\sqrt{2}}}(A+B)g(N\wedge u_2,X_k),\hfill \\ 0\hfill & =g(R(X_1,X_3)N,X_k)=\epsilon \mu Bg(N,u_2)g(N\wedge u_2,X_k)=\epsilon \mu {\displaystyle \frac{B}{\sqrt{2}}}(A+B)g(N\wedge u_2,X_k),\hfill \\ 0\hfill & =g(R(X_2,X_3)N,X_k)=\epsilon \mu Cg(N,u_2)g(N\wedge u_2,X_k)=\epsilon \mu {\displaystyle \frac{C}{\sqrt{2}}}(A+B)g(N\wedge u_2,X_k).\hfill \end{array}$$

(18)

It is easy to check that $g(N\wedge u_2,X_k)=0$ for every $k\in \{1,2,3\}$ if and only if either N vanishes or it is lightlike. As a consequence, taking into account the fact that $A,B,C$ cannot all vanish, conditions (18) hold if and only if $A+B=0$; whence, $N=A(E_1-E_2)+C{E}_3=\sqrt{2}A{u}_2+C{u}_3$. Since $\delta =C^2$, then $\delta =1$. Thus, $\mathsf{\Sigma }$ is timelike. Without the loss of generality, we can fix $C=1$. By setting $\varphi =\sqrt{2}A$, we obtain the result (I) given in the statement.

Next, let $\mathsf{\Sigma }$ be any Codazzi surface in $\mathcal{E}$, as described in (I). Thus, the vector fields

$$X=u_2,Y=u_1+u_2-\varphi u_3$$

(19)

span the tangent space to $\mathsf{\Sigma }$ at every point.

We observe that this distribution is integrable if and only if $X\left(\varphi \right)=-\mu$. In fact, we have

$$[X,Y]=-(\mu +X(\varphi \left)\right)N+\varphi X\left(\varphi \right)X.$$

Moreover, a direct calculation, using the Gauss equation, (8) and (19), allows us to obtain

$${\nabla }_X^{\mathsf{\Sigma }}X={\nabla }_Y^{\mathsf{\Sigma }}X=0,{\nabla }_X^{\mathsf{\Sigma }}Y=-\mu \varphi X,{\nabla }_Y^{\mathsf{\Sigma }}Y=\varphi Y\left(\varphi \right)X$$

(20)

and the second fundamental form h of the immersion is determined by

$$\begin{array}{cc}h(X,X)=0,\hfill & h(X,Y)=-X\left(\varphi \right)-{\displaystyle \frac{\mu }{2}},\hfill \\ h(Y,X)={\displaystyle \frac{\mu }{2}},\hfill & h(Y,Y)=\epsilon -Y\left(\varphi \right).\hfill \end{array}$$

(21)

It easily follows from (20) that $R^{\mathsf{\Sigma }}=0$; therefore, any Codazzi surface is flat, as stated in point (II).

Next, Equation (21) yields that $h(Y,X)={\displaystyle \frac{\mu }{2}}\ne 0$ (hence, $\mathsf{\Sigma }$ cannot be totally geodesic) and the symmetry condition for h is equivalent to the integrability condition $X\left(\varphi \right)=-\mu$ for the distribution $\{X,Y\}$. Since $\mu \ne 0$, it excludes that $X\left(\varphi \right)$ vanishes. In particular, we have $\varphi \ne 0$.

From (21), a standard calculation shows that the mean curvature of $\mathsf{\Sigma }$ is given by

$$H={\displaystyle \frac{1}{2}}{\mathrm{tr}}_{g^{\mathsf{\Sigma }}}h={\displaystyle \frac{1}{2}}\sum {\left(g^{\mathsf{\Sigma }}\right)}^{ij}{h}_{ij}={\displaystyle \frac{\mu }{2}},$$

which proves the point (III) of the statement, i.e., the fact that any Codazzi surface is CMC but never minimal.

Finally, in order to consider the special case of parallel surfaces, we shall make use of the results given in Theorem 1. As previously emphasized, $\mathcal{E}$ is the special case corresponding to $\alpha =\mu \ne 0$ and $\beta -\eta =-\mu \ne 0$. As $\alpha \ne 0$, this excludes case (a) of Theorem 1. On the other hand, case (b) cannot occur for $\mathcal{E}$, either. In fact, in such a case, we have a totally geodesic surface; however, we already proved that totally geodesic surfaces do not exist in $\mathcal{E}$. Therefore, as stated in point (IV), $\mathcal{E}$ does not admit any parallel surfaces, and this completes the proof. □

Next, we shall exhibit a class of examples of Codazzi surfaces, which, by the above Theorem 2, are CMC and flat.

Example 1.

From Theorem 2, the normal vector field associated with any Codazzi surface is of the form $N=\varphi u_2+u_3$. In general, $\varphi =\varphi (u,v)$ is a smooth function defined on the surface Σ, which depends on some local coordinates $(u,v)$ defined on Σ, if they exist.

Using the tangent vector fields described in (19), we find that

$$\begin{array}{c}\hfill {\partial }_u=\beta (u,v)X=\beta (u,v)u_2,{\partial }_v=Y=u_1+u_2-\varphi (u,v)u_3,\end{array}$$

(22)

are coordinate vector fields, provided that the smooth function β satisfies

$${\beta }_v(u,v)=-\mu \varphi (u,v)\beta (u,v),\beta (u,v)=-{\displaystyle \frac{{\varphi }_u(u,v)}{\mu }}.$$

(23)

We now show that the plane Σ, defined by the equation $x_3=0$, admits some coordinates as described in (22) and (23). Therefore, Σ provides an example of a Codazzi surface (whence, a flat, CMC surface) for any choice of ϕ.

Let $F:\mathsf{\Sigma }\to \mathcal{E},(u,v)\mapsto (x_1(u,v),x_2(u,v),x_3(u,v))$ be the immersion of the surface Σ in the local coordinates in (22). By combining (7) and (22) with $x_3=0$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}{\partial }_u{x}_1(u,v)=0,\hfill & {\partial }_v{x}_1(u,v)=1,\hfill \\ {\partial }_u{x}_2(u,v)=\beta (u,v),\hfill & {\partial }_v{x}_2(u,v)={\displaystyle \frac{{\mu }^2{x}_2^2(u,v)}{2}}+1,\hfill \\ {\partial }_u{x}_3(u,v)=0,\hfill & {\partial }_v{x}_3(u,v)=-\mu x_2(u,v)-\varphi (u,v).\hfill \end{array}\end{array}$$

(24)

By integration of (24), we deduce

$$x_1(u,v)=v+c_1,x_2(u,v)={\displaystyle \frac{\sqrt{2}}{\mu }}tan\left({\displaystyle \frac{\mu }{\sqrt{2}}}v+f\left(u\right)\right),x_3(u,v)=0$$

for some smooth function f, which depends only on u.

Moreover, as $x_3=0$, we have ${\partial }_v{x}_3(u,v)=0$; therefore, using the last equation in (24),

$$x_2(u,v)=-{\displaystyle \frac{\varphi (u,v)}{\mu }}.$$

(25)

It is now easy to check that condition (23) is satisfied for any suitable choice of $f\left(u\right)$ that determines ϕ by (25) and then β as $\beta (u,v)=-{\displaystyle \frac{{\varphi }_u(u,v)}{\mu }}$. Observe that since $\beta \ne 0$, we have ${\varphi }_u\ne 0$, i.e., $f^{\prime }\left(u\right)\ne 0$, so that f is not constant.

Finally, since translations in $x_1$ are isometries of the ambient space, we conclude that the plane Σ defined by $x_3=0$ can be parametrized by the following CMC immersion:

$$F(u,v)=\left(v,{\displaystyle \frac{\sqrt{2}}{\mu }}tan\left({\displaystyle \frac{\mu }{\sqrt{2}}}v+f\left(u\right)\right),0\right)$$

for any arbitrary non-constant smooth function $f\left(u\right)$.

4. Totally Umbilical Surfaces in $\mathcal{E}$

In this section, we will investigate the existence of totally umbilical surfaces in the exceptional example $\mathcal{E}$. Let $\mathsf{\Sigma }$ denote any oriented surface immersed into $\mathcal{E}$, and let N be its $\delta -$unit normal vector field ($g(N,N)=\delta$). The vector field $u_2$ decomposes as

$$u_2=T+\nu N,$$

where $\nu :=\delta g(N,u_2)$, and T is tangent to $\mathsf{\Sigma }$; whence,

$$g(T,T)=-\delta {\nu }^2.$$

Remark 1. 

We observe that $\nu =0$ if and only if $g(N,u_2)=0$, i.e., the surface Σ is Codazzi (see Theorem 2). Then, in the rest of this section, we will consider two cases separately, depending on whether ν vanishes. We explicitly observe that from Theorem 2, totally umbilical surfaces Σ cannot be totally geodesic.

We now prove the following:

Proposition 2. 

Any totally umbilical surface Σ immersed in $\mathcal{E}$ has a Codazzi second fundamental form.

Proof. 

If $\mathsf{\Sigma }$ is a surface, immersed in $\mathcal{E}$, which is not Codazzi, then $\nu \ne 0$. Assume now that $\mathsf{\Sigma }$ is totally umbilical, with a unit normal N such that $g(N,N)=\delta$.

Observe that if T vanishes identically on a nonempty open subset, we can choose $N=u_2$ on this set, which contradicts the fact that, as $g(N,N)=\delta$, N is not lightlike. Therefore, we can assume that T does not vanish on any nonempty open subset.

Next, we define the vector field $JX:=N\wedge X$, as some sort of rotation of X around an axis that follows the direction of N and satisfies

$$g\left(JX,JY\right)=-\delta g(X,Y),J^{2}X=-\delta X,$$

for any vector fields $X,Y$ tangent to $\mathsf{\Sigma }$. As a consequence, $JT=N\wedge (u_2-\nu N)=N\wedge u_2$ is tangent to $\mathsf{\Sigma }$ and horizontal.

We recall that $\mathsf{\Sigma }$ is totally umbilical if and only if ${\nabla }_{X}N=\lambda X$ for every X tangent to $\mathsf{\Sigma }$. We can now calculate

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\nabla }_{T}JT=& {\nabla }_T(N\wedge T)\hfill \\ \hfill =& \left({\nabla }_{T}N\right)\wedge T+N\wedge \left({\nabla }_{T}T\right)\hfill \\ \hfill =& -\lambda T\wedge T+N\wedge \left({\nabla }_T(u_2-\nu N)\right)\hfill \\ \hfill =& -{\displaystyle \frac{\mu }{2}}N\wedge (T\wedge u_2)+\lambda \nu N\wedge T\hfill \\ \hfill =& -{\displaystyle \frac{\mu }{2}}\delta \nu T+\lambda \nu JT,\hfill \\ \hfill {\nabla }_{JT}T=& {\nabla }_{JT}(u_2-\nu N)\hfill \\ \hfill =& -{\displaystyle \frac{\mu }{2}}(JT\wedge u_2)-JT\left(\nu \right)N+\nu \lambda JT\hfill \\ \hfill =& -{\displaystyle \frac{\mu }{2}}(JT\wedge T)-{\displaystyle \frac{\mu }{2}}\nu (JT\wedge N)-JT\left(\nu \right)N+\nu \lambda JT\hfill \\ \hfill =& -{\displaystyle \frac{\mu }{2}}\delta \nu T+\lambda \nu JT,\hfill \end{array}\end{array}$$

where we used

$$\begin{array}{cc}\hfill JT\left(\nu \right)N=& \delta JT\left(g(N,u_2)\right)N\hfill \\ \hfill =& \delta \left(g({\nabla }_{JT}N,u_2)+g(N,{\nabla }_{JT}{u}_2)\right)N\hfill \\ \hfill =& \delta \left(-\lambda g(JT,T+\nu N)-{\displaystyle \frac{\mu }{2}}g(N,JT\wedge (T+\nu N))\right)N\hfill \\ \hfill =& -\delta {\displaystyle \frac{\mu }{2}}g(N,JT\wedge T)N=-{\displaystyle \frac{\mu }{2}}JT\wedge T.\hfill \end{array}$$

Moreover, we obtain

$$\begin{array}{cc}\hfill {\nabla }_{JT}JT=& {\nabla }_{JT}(N\wedge T)\hfill \\ \hfill =& \left({\nabla }_{JT}N\right)\wedge T+N\wedge \left({\nabla }_{JT}T\right)\hfill \\ \hfill =& -\lambda (JT\wedge T)-\delta {\displaystyle \frac{\mu }{2}}\nu JT-\delta \nu \lambda T.\hfill \end{array}$$

Therefore, $h(JT,JT)=-\delta \lambda g(JT\wedge T,N)$. If we require total umbilicity, we then get

$$JT\left(\nu \right){\displaystyle \frac{2\lambda }{\mu }}=-\delta \lambda g(JT\wedge T,N)=\lambda g(JT,JT)=\lambda {\nu }^2.$$

However, $\lambda \ne 0$ because $\mathsf{\Sigma }$ cannot be totally geodesic according to Theorem 2. Therefore, the above equation yields

$$JT\left(\nu \right)={\displaystyle \frac{\mu }{2}}{\nu }^2.$$

(26)

Observe that $JT\left(\nu \right)\ne 0$ as $\nu \ne 0$. Moreover, as ${\nabla }_{T}JT={\nabla }_{JT}T$, we obtain that $[T,JT]=0$. Therefore,

$$\begin{array}{cc}\hfill R(T,JT)N& =-{\nabla }_T{\nabla }_{JT}N+{\nabla }_{JT}{\nabla }_{T}N\hfill \\ \hfill & ={\nabla }_T\lambda JT-{\nabla }_{JT}\lambda T\hfill \\ \hfill & =T\left(\lambda \right)JT-JT\left(\lambda \right)T.\hfill \end{array}$$

(27)

On the other hand, by (16), we obtain

$$R(T,JT)N=\epsilon \mu \nu g(T\wedge JT,N)JT=2\epsilon \mu \delta JT\left(\nu \right)JT.$$

(28)

We now compare (27) and (28), and we get

$$T\left(\lambda \right)=2\epsilon \mu \delta JT\left(\nu \right),JT\left(\lambda \right)=0,$$

whence, using $[T,JT]\left(\lambda \right)=0$, we deduce

$$2\epsilon \mu \delta JT\left(JT\right(\nu \left)\right)=JT\left(T\right(\lambda \left)\right)=T\left(JT\right(\lambda \left)\right)=0.$$

Consequently, Equation (26) yields

$$\begin{array}{c}\hfill \begin{array}{c}\hfill 0=JT\left(JT\right(\nu \left)\right)=\mu \nu JT\left(\nu \right),\end{array}\end{array}$$

which cannot occur, since $\nu \ne 0$ and $JT\left(\nu \right)\ne 0$. □

Next, we are left to investigate the existence of totally umbilical surfaces in the class of Codazzi surfaces.

Let $\mathsf{\Sigma }$ be a Codazzi surface immersed in $\mathcal{E}$. We know from Theorem 2 that its normal vector field is given by $N=\varphi u_2+u_3$. We recall that in this case, vector fields $X,Y$ described in (19) span the tangent space to $\mathsf{\Sigma }$.

From (21), we deduce that $\mathsf{\Sigma }$ is totally umbilical if and only if

$$\left\{\begin{array}{c}{\displaystyle \frac{\mu }{2}}=h(X,Y)=\lambda g(X,Y)=\lambda \hfill \\ \epsilon -Y\left(\varphi \right)=h(Y,Y)=\lambda g(Y,Y)=\lambda (2+{\varphi }^2).\hfill \end{array}\right.$$

Next, we find

$$h(X,X)=0,h(X,Y)={\displaystyle \frac{\mu }{2}},h(Y,Y)={\displaystyle \frac{\mu }{2}}{\varphi }^2+\mu .$$

(29)

Therefore, using (20) and (29), we obtain

$$\begin{array}{c}{\nabla }_X^{\mathsf{\Sigma }}h(X,X)={\nabla }_Y^{\mathsf{\Sigma }}h(X,X)=0,\hfill \\ {\nabla }_X^{\mathsf{\Sigma }}h(X,Y)={\nabla }_Y^{\mathsf{\Sigma }}h(X,Y)=0,\hfill \\ {\nabla }_X^{\mathsf{\Sigma }}h(Y,Y)=\mu \varphi X\left(\varphi \right)+2\mu \varphi h(X,Y)=0,\hfill \\ {\nabla }_Y^{\mathsf{\Sigma }}h(Y,Y)=\mu \varphi Y\left(\varphi \right)-2\varphi Y\left(\varphi \right)h(X,Y)=0.\hfill \end{array}$$

This proves that a totally umbilical surface in $\mathcal{E}$ is necessarily parallel. Then, taking into account Theorem 2 and Proposition 2, we obtain the following conclusion.

Theorem 3. 

The exceptional examples $\mathcal{E}$ do not admit any totally umbilical surface.

As we already mentioned in the Introduction Section, totally umbilical and parallel (in particular, totally geodesic) surfaces were investigated in the Lorentzian Heisenberg groups [31], Lorentzian BCV spaces ${\mathbb{M}}_1^3(\kappa ,\tau )$ with $\tau \ne 0$[8], Lorentzian reducible spaces [35] (which in the homogeneous case correspond to ${\mathbb{M}}_1^3(\kappa ,0)$)-spaces), and homogeneous plane waves $\mathcal{H}$ [34]. Therefore, the present work completed the investigation of totally umbilical and parallel surfaces of three-dimensional homogeneous Lorentzian manifolds with a four-dimensional isometry group. The results concerning the existence of these surfaces in all such homogeneous Lorentzian manifolds are summarized in Table 1 below.

Table 1. Surfaces in 3D Lorentzian spaces with 4D isometry group.

	${\mathbb{M}}_1^3(\mathit{\kappa },\mathit{\tau }\ne 0)$	${\mathbb{M}}_1^3(\mathit{\kappa },0)$	$\mathcal{E}$	$\mathcal{H}$
Totally Umbilical	✗	✓	✗	✗
Totally Geodesic	✗	✓	✗	✓
Parallel	✓	✓	✗	✓