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Geometric Consequences of Four Dimensional Neutral Lie Groups

  • Amirhesam ZaeimEmail author
  • Ramisa Karami
Article

Abstract

Geometry of four dimensional pseudo-Riemannian Lie groups of signature (2, 2) studied. A rich family of Einstein, locally symmetric and conformally flat examples is presented.

Keywords

Lie group Neutral signature Symmetric space 

Mathematics Subject Classification.

53C50 53C30 53C35 

1 Introduction

Lie groups which are the most famous category through the homogeneous spaces are so interesting in the filed of mathematics and physics. Generally, a homogeneous manifold is identified by the action of it’s isometry group. In fact, the manifold (Mg) is homogeneous if and only if I(M) acts transitively on M. In this case, (Mg) could be presented as a quotient of the Lie groups G / H, with an invariant metric g (O’Neill 1983).

Four dimensional simply connected Riemannian Lie groups studied in Arias-Marco and Kowalski (2008), but Bérard-Bérgery (1985) considered the general case and showed that a simply connected four dimensional homogeneous Riemannian manifolds is either symmetric or isometric to a Lie groups equipped with a left-invariant Riemannian metric. Up to recent, most of investigations in the field of homogeneous manifolds were focused on the Riemannian case.

The pseudo-Riemannian counterpart of the results of Tricerri and Vanhecke (1983) studied in Calvaruso (2007), were the author studied three-dimensional homogeneous Lorentzian manifolds and proved that a connected, simply connected and complete homogeneous Lorentzian manifold is either symmetric or isometric to a three dimensional Lorentzian Lie group (Mg), equipped with a left-invariant metric g. Specially, the Lorentzian Lie groups of dimension three, classified in Cordero and Parker (1997) and Rahmani (1992). A full classification of four dimensional homogeneous manifolds with non-trivial isotropy presented in Komrakov (2001), but this study leaves empty the place of Lie groups through the classification of homogeneous spaces. Based to this classification, deep investigations of the homogeneous four dimensional manifolds with non-trivial isotropy were done (see for example Haji-Badali and Zaeim 2015; Zaeim 2017; Zaeim and Haji-Badali 2016).

Considering curvature properties on different geometric spaces is of special interest. A complete classification of four-dimensional Einstein-like Lorentzian Lie groups were considered in Zaeim (2017). Curvature conditions on four-dimensional Lorentzian and neutral Lie groups, with special motivation to Einstein and Ricci parallel examples, studied in Calvaruso and Zaeim (2013, 2015) respectively, but the problem of considering geometric consequence of neutral Lie groups is still open. In this paper, in order to complete study of curvature properties in four dimensional neural Lie groups, we bring a full classification of these spaces.

This paper is organized in the following way. Some basic facts about four dimensional neutral Lie groups brought to the Sect. 2. This section also contains a full classification of four dimensional neutral Lie groups, where the restriction of the left invariant metric on a three dimensional subgroup is Lorentzian or degenerate (referring to Haji-Badali and Karami 2017). Finally, the geometry of each class is studied in the last section.

2 Neutral Lie Groups of Dimension Four

Study of homogenous Riemannian manifolds of dimension four by Bŕard-Bérgery in Bérard-Bérgery (1985), highlighted the important role of Riemannian Lie groups. By this research, any homogeneous Riemannian manifold of dimension four is either symmetric or isometric to a Lie group, equipped with a left-invariant Riemannian metric. Arias-Marco and Kowalski (2008) classified four-dimensional Riemannian Lie groups. We resume this classification in the following proposition.

Proposition 2.1

(Arias-Marco and Kowalski 2008) A simply connected four-dimensional Riemannian Lie group is : 
  • (i) either one of the unsolvable direct products \( SU(2) \times \mathbb {R}\) and \(\widetilde{SL}(2,\mathbb {R})\times \mathbb {R};\) or

  • (ii) one of the following solvable Lie groups : 
    • (ii1) the non-trivial semi-direct products \(E(2)\rtimes \mathbb {R}\) and \(E(1,1)\rtimes \mathbb {R};\)

    • (ii2) the non-nilpotent semi-direct products \(H\rtimes \mathbb {R},\) where H denotes the Heisenberg group; 

    • (ii3) the semi-direct products \(\mathbb {R}^3\rtimes \mathbb {R}.\)

Following the work (Calvaruso and Zaeim 2015), if G is a four-dimensional simply connected Lie group, equipped with a left invariant metric of neutral signature, then G is one of Lie groups listed in the above Proposition 2.1. Now, let \((G=G_3\rtimes \mathbb {R},g)\) is a pseudo-Riemannian Lie group of dimension four, where \(G_3\) is one of the Lie groups SU(2), \(\widetilde{SL}(2,\mathbb {R})\), E(2), E(1, 1), H or \(\mathbb {R}^3\), and we show the Lie algebra of \(G_3\) by \({\mathfrak {g}}_3\). Based on the study (Calvaruso and Zaeim 2015), authors in Haji-Badali and Karami (2017) categorized four dimensional neutral Lie groups in the following two cases
  1. (a)

    \(g|_{{\mathfrak {g}}_3}\) is Lorentzian and the time-like vector \(e_4\) acts as a derivation on \({\mathfrak {g}}_3\),

     
  2. (b)

    \(g|_{{\mathfrak {g}}_3}\) is degenerate and the light-like vector \(e_4\) acts as a derivation on \({\mathfrak {g}}_3\),

     
and could classify four dimensional neutral Lie groups in two theorems which we report this classification in here.

Case (a): Restricting the metric on \({\mathfrak {g}}_3\) is Lorentzian.

A classification of three dimensional homogeneous Lorentzian manifolds were done in Calvaruso (2007). This context contains a full classification of three dimensional Lorentzian Lie groups through seven classes, named \({\mathfrak {g}}1, \ldots , {\mathfrak {g}}7\), cases correspond to unimodular and non-unimodular Lie groups. Following this classification, we have,

Theorem 2.2

(Haji-Badali and Karami 2017) Let \((G=G_3\rtimes \mathbb {R},g)\) be a four dimensional Lie group of neutral signature,  where the restriction of g on \({\mathfrak {g}}_3\) is Lorentzian. In this case,  there exist a pseudo-orthonormal basis such that \({\mathfrak {g}}\) is isometric to one of the following cases : 
  1. (a)
    \({\mathfrak {g}}_3\) is of type \({\mathfrak {g}}1\)
    1.

    \({\begin{array}{lll} \left[ e_1,e_2\right] =\alpha e_1,&{}\left[ e_1,e_3\right] =-\alpha e_1,&{}\left[ e_2,e_3\right] =\alpha e_2+\alpha e_3,\\ \left[ e_1,e_4\right] =Ae_1,&{}\left[ e_2,e_4\right] =Be_1+Ce_2+Ce_3, &{}\left[ e_3,e_4\right] =-Be_1+De_2+De_3,\quad \alpha \ne 0, \end{array}}\)

    2.

    Open image in new window

     
  2. (b)
    \({\mathfrak {g}}_3\) is of type \({\mathfrak {g}}2\)
    1.

    \(\begin{array}{lll} \left[ e_1,e_2\right] =-\gamma e_2,&{}\left[ e_1,e_3\right] =\gamma e_3,&{} \left[ e_1,e_4\right] =Ae_2+Be_3,\\ \left[ e_2,e_4\right] =Ce_2,&{}\left[ e_3,e_4\right] =De_3, &{}\gamma \ne 0, \end{array}\)

    2.

    \(\begin{array}{ll} \left[ e_1,e_2\right] =-\gamma e_2-\beta e_3,&{}\left[ e_1,e_3\right] =-\beta e_2+\gamma e_3,\\ \left[ e_1,e_4\right] =Ae_2+Be_3,&{}\left[ e_2,e_4\right] =Ce_2+De_3,\\ \left[ e_3,e_4\right] =De_2-\frac{2\gamma D-\beta C}{\beta }e_3,&{}\beta \gamma \ne 0, \end{array}\)

    3.

    \({\begin{array}{ll} \left[ e_1,e_2\right] =-\gamma e_2-\beta e_3,&{}\left[ e_1,e_3\right] =-\beta e_2+\gamma e_3,\\ \left[ e_2,e_3\right] =\alpha e_1,&{}\left[ e_1,e_4\right] =\frac{(\gamma ^2+\beta ^2)A-\beta \alpha B}{\alpha \gamma }e_2+Be_3,\\ \left[ e_2,e_4\right] =\frac{\alpha B-\beta A}{\gamma }e_1-Ce_2-\frac{\beta C}{\gamma }e_3,&{}\left[ e_3,e_4\right] =Ae_1-\frac{\beta C}{\gamma }e_2+Ce_3,\quad \alpha \gamma \ne 0. \end{array}}\)

     
  3. (c)
    \({\mathfrak {g}}_3\) is of type \({\mathfrak {g}}3\)
    1.

    \({\begin{array}{lll} \left[ e_1,e_2\right] =-\gamma e_3,&{} [e_1,e_3]=-\beta e_2,&{}\left[ e_2,e_3\right] =\alpha e_1,\\ \left[ e_1,e_4\right] =-\frac{\beta A}{\alpha } e_2+\frac{\gamma B}{\alpha } e_3,&{}\left[ e_2,e_4\right] =Ae_1+\frac{\gamma C}{\beta }e_3,&{} [e_3,e_4]=Be_1+Ce_2,\quad \alpha \beta \ne 0, \end{array}}\)

    2.

    \({\begin{array}{lll} \left[ e_1,e_2\right] =-\gamma e_3,&{} [e_2,e_3]=\alpha e_1,\\ \left[ e_1,e_4\right] =A e_1+\frac{\gamma B}{\alpha }e_3,&{}\left[ e_2,e_4\right] =Ce_1+De_3,&{} [e_3,e_4]=Be_1+Ae_3,\quad \alpha \ne 0, \end{array}}\)

    3.

    \({\begin{array}{lll} \left[ e_1,e_2\right] =-\gamma e_3,&{} [e_1,e_3]=-\beta e_2,\\ \left[ e_1,e_4\right] =Ae_2+Be_3,&{} [e_2,e_4]=De_2+\frac{\gamma C}{\beta } e_3,&{} [e_3,e_4]=Ce_2+De_3,\quad \beta \ne 0, \end{array}}\)

    4.

    \({\begin{array}{lll} \left[ e_1,e_3\right] =-\beta e_2,&{} [e_2,e_3]=\alpha e_1,\\ \left[ e_1,e_4\right] =Ae_1-\frac{\beta B}{\alpha }e_2,&{} [e_2,e_4]=Be_1+Ae_2,&{} [e_3,e_4]=Ce_1+De_2,\quad \alpha \ne 0, \end{array}}\)

    5.

    \(\begin{array}{ll} \left[ e_1,e_2\right] =-\gamma e_3,&{}\left[ e_1,e_4\right] =(A-B)e_1+Ce_2+De_3\\ \left[ e_2,e_4\right] =Ee_1+Be_2+Fe_3,&{}\left[ e_3,e_4\right] =Ae_3,\quad \gamma \ne 0, \end{array}\)

    6.

    \(\begin{array}{ll} \left[ e_1,e_3\right] =-\beta e_2,&{} \left[ e_1,e_4\right] =(A-B)e_1+Ce_2+De_3\\ \left[ e_2,e_4\right] =Ae_2,&{} [e_3,e_4]=Ee_1+Fe_2+Be_3,\quad \beta \ne 0, \end{array}\)

    7.

    \(\begin{array}{ll} \left[ e_2,e_3\right] =\alpha e_1,&{}\left[ e_1,e_4\right] =(A+B)e_1\\ \left[ e_2,e_4\right] =Ce_1+Ae_2+De_3,&{} [e_3,e_4]=Ee_1+Fe_2+Be_3,\quad \alpha \ne 0, \end{array}\)

    8.

    \({\begin{array}{lll} \left[ e_1,e_4\right] =Ae_1+Be_2+Ce_3,&\left[ e_2,e_4\right] =De_1+Ee_2+Fe_3,&\left[ e_3,e_4\right] =Ge_1+He_2+Ke_3. \end{array}}\)

     
  4. (d)
    \({\mathfrak {g}}_3\) is of type \({\mathfrak {g}}4\)
    1.

    \(\begin{array}{ll} \left[ e_1,e_2\right] =-e_2+(2\varepsilon -\beta ) e_3,&{}\left[ e_1,e_3\right] =-\beta e_2+e_3,\\ \left[ e_1,e_4\right] =Ae_2+Be_3,&{}\left[ e_2,e_4\right] =Ce_2+(D-C)(\varepsilon -\frac{\beta }{2})e_3,\\ \left[ e_3,e_4\right] =\frac{\beta }{2}(C-D)e_2+De_3,&{}\varepsilon ^2=1, \end{array}\)

    2.

    \({\begin{array}{ll} \left[ e_1,e_2\right] =-e_2+\varepsilon e_3,&{}\left[ e_1,e_3\right] =-\varepsilon e_2+e_3,\\ \left[ e_1,e_4\right] =Ae_1+Be_2+Ce_3,&{}\left[ e_2,e_4\right] =De_1+Ee_2-\frac{\varepsilon }{2}(A+E-F)e_3,\\ \left[ e_3,e_4\right] =\varepsilon De_1-\frac{\varepsilon }{2}(A-E+F)e_2+Fe_3,&{}\varepsilon ^2=1, \end{array}}\)

    3.

    \({\begin{array}{ll} \left[ e_1,e_2\right] =-e_2+(2\varepsilon -\beta ) e_3,&{}\left[ e_1,e_3\right] =-\beta e_2+e_3,\\ \left[ e_2,e_3\right] =\alpha e_1,&{}\left[ e_1,e_4\right] =Ae_2-\frac{(2\varepsilon -\beta )(\beta B+\alpha A)-B}{\alpha }e_3,\\ \left[ e_2,e_4\right] =Be_1-Ce_2+(2\varepsilon -\beta )Ce_3,&{}\left[ e_3,e_4\right] =(B \beta +A\alpha )e_1-C\beta e_2+Ce_3,\ \alpha \ne 0,\ \varepsilon ^2=1, \end{array}}\)

    4.

    \({\begin{array}{ll} \left[ e_1,e_2\right] =-e_2+\varepsilon e_3,&{}\left[ e_1,e_3\right] =-\varepsilon e_2+e_3,\\ \left[ e_1,e_4\right] =(A+B)e_1-\varepsilon Ce_2+Ce_3,&{}\left[ e_2,e_3\right] =\alpha e_1,\\ \left[ e_2,e_4\right] =De_1+Ae_2-\varepsilon Ae_3,&{}\left[ e_3,e_4\right] =\varepsilon (D-\alpha C)e_1-\varepsilon Be_2+Be_3,\quad \varepsilon ^2=1. \end{array}}\)

     
  5. (e)
    \({\mathfrak {g}}_3\) is of type \({\mathfrak {g}}5\)
    1.

    \(\begin{array}{ll} \left[ e_1,e_3\right] =\alpha e_1+\beta e_2,&{}\left[ e_2,e_3\right] =\gamma e_1+\delta e_2,\\ \left[ e_1,e_4\right] =(B+\frac{A}{\gamma }(\alpha -\delta ))e_1+\frac{\beta A}{\gamma }e_2,&{}\left[ e_2,e_4\right] =Ae_1+Be_2,\\ \left[ e_3,e_4\right] =Ce_1+De_2,&{} \alpha +\delta \ne 0,\ \alpha \gamma +\beta \delta =0,\ \gamma \ne 0, \end{array}\)

    2.

    \(\begin{array}{lll} \left[ e_1,e_3\right] =\alpha e_1,&{}\left[ e_2,e_3\right] =\delta e_2,\\ \left[ e_1,e_4\right] =Ae_1,&{}\left[ e_2,e_4\right] =Be_2,&{}\left[ e_3,e_4\right] =Ce_1+De_2,\quad \alpha +\delta \ne 0, \end{array}\)

    3.

    \({\begin{array}{lll} \left[ e_1,e_3\right] =\alpha e_1,&{}\left[ e_2,e_3\right] =\alpha e_2,\\ \left[ e_1,e_4\right] =Ae_1+Be_2,&{}\left[ e_2,e_4\right] =Ce_1+De_2,&{}\left[ e_3,e_4\right] =Ee_1+Fe_2,\quad \alpha \ne 0, \end{array}}\)

    4.

    \({\begin{array}{lll} \left[ e_1,e_3\right] =\alpha e_1+\beta e_2,\\ \left[ e_1,e_4\right] =Ae_1+\frac{\beta (A-B)}{\alpha }e_2,&{}\left[ e_2,e_4\right] =Be_2,&{}\left[ e_3,e_4\right] =Ce_1+De_2,\quad \alpha \ne 0. \end{array}}\)

     
  6. (f)
    \({\mathfrak {g}}_3\) is of type \({\mathfrak {g}}6\)
    1.

    \(\begin{array}{ll} \left[ e_1,e_2\right] =\alpha e_2+\beta e_3,&{}\left[ e_1,e_3\right] =\gamma e_2+\delta e_3,\\ \left[ e_1,e_4\right] =Ae_2+Be_3,&{}\left[ e_2,e_4\right] =(D+\frac{C}{\gamma }(\alpha -\delta ))e_2+\frac{\beta C}{\gamma }e_3,\\ \left[ e_3,e_4\right] =Ce_2+De_3,&{} \alpha +\delta \ne 0,\ \alpha \gamma -\beta \delta =0,\ \gamma \ne 0, \end{array}\)

    2.

    \(\begin{array}{lll} \left[ e_1,e_2\right] =\alpha e_2,&{}\left[ e_1,e_3\right] =\delta e_3,\\ \left[ e_1,e_4\right] =Ae_2+Be_3,&{}\left[ e_2,e_4\right] =Ce_2,&{}\left[ e_3,e_4\right] =De_3,\quad \alpha +\delta \ne 0, \end{array}\)

    3.

    \({\begin{array}{lll} \left[ e_1,e_2\right] = \alpha e_2,&{}\left[ e_1,e_3\right] =\alpha e_3,\\ \left[ e_1,e_4\right] =Ae_2+Be_3,&{}\left[ e_2,e_4\right] =Ce_2+De_3,&{}\left[ e_3,e_4\right] =Ee_2+Fe_3,\quad \alpha \ne 0, \end{array}}\)

    4.

    \(\begin{array}{ll} \left[ e_1,e_2\right] =\alpha e_2+\beta e_3,&{}\left[ e_1,e_4\right] =Ae_2+Be_3,\\ \left[ e_2,e_4\right] =Ce_2+\frac{\beta (C-D)}{\alpha }e_3,&{}\left[ e_3,e_4\right] =De_3,\quad \alpha \ne 0. \end{array}\)

     
  7. (g)
    \({\mathfrak {g}}_3\) is of type \({\mathfrak {g}}7\)
    1.

    \({\begin{array}{ll} \left[ e_1,e_2\right] =-\beta e_2-\beta e_3,&{} \left[ e_1,e_3\right] =\beta e_2+\beta e_3,\\ \left[ e_2,e_3\right] =\gamma e_1+\delta e_2+\delta e_3,&{} \left[ e_1,e_4\right] =Ae_1-\frac{\beta (A-B-C)}{\delta } e_2-\frac{\beta (A-B-C)}{\delta } e_3,\\ \left[ e_2,e_4\right] =-\frac{\gamma (A-B-C)+\delta D}{\delta } e_1+B e_2+B e_3,&{} \left[ e_3,e_4\right] =De_1+Ce_2+Ce_3,\quad \delta \ne 0. \end{array}}\)

    2.

    \({\begin{array}{ll} \left[ e_1,e_2\right] =-\alpha e_1-\beta e_2-\beta e_3,&{} [e_1,e_3]=\alpha e_1+\beta e_2+\beta e_3,\\ \left[ e_1,e_4\right] =Ae_1+\frac{\beta (A-B-C)}{\alpha }e_2+\frac{\beta (A-B-C)}{\alpha }e_3,&{}[e_2,e_4]=-De_1+Be_2+Be_3,\\ \left[ e_3,e_4\right] =De_1+Ce_2+Ce_3,\quad \alpha \ne 0. \end{array}}\)

     

Case (b): Restricting the metric on \({\mathfrak {g}}_3\) is degenerate.

Theorem 2.3

(Haji-Badali and Karami 2017) Let \((G=G_3\rtimes \mathbb {R},g)\) be a four dimensional Lie group of neutral signature,  when the restriction of g on \({\mathfrak {g}}_3\) is degenerate. In this case,  one of the following cases may happen : 
  1. (a)
    There exist a basis \(\{e_1,\dots ,e_4\}\) for \({\mathfrak {g}}\) such that \({\mathfrak {g}}_3=\mathrm{Span}(e_1,e_2,e_3)\) and \(\mathfrak {r}=\mathrm{Span}(e_4)\) and the non-zero inner products is determined by \(g(e_1,e_1)=-g(e_2,e_2)=g(e_3,e_4)=g(e_4,e_3)=1,\) where the non-zero Lie brackets are : 
    1. (a1)
      \({\mathfrak {g}}'_3\)is trivial: In this case \({\mathfrak {g}}=\mathfrak {r}^3\rtimes \mathfrak {r}\) is isometric the following Lie algebra:
      $$\begin{aligned}&\left[ e_1,e_4\right] =Ae_1+B e_2+C e_3, \left[ e_2,e_4\right] =D e_1+E e_2+Fe_3, \left[ e_3,e_4\right] \\&\quad =Ge_1+He_2+Ke_3. \end{aligned}$$
       
    2. (a2)
      \({\mathfrak {g}}'_3\)is one dimensional: In this case \({\mathfrak {g}}={\mathfrak {h}}\rtimes \mathfrak {r}\) is isometric to one of the following Lie algebras, where \({\mathfrak {h}}\) is the three dimensional Heisenberg Lie algebra:
      1.

      Open image in new window

      2.

      \(\begin{array}{lll} \left[ e_1,e_3\right] =Ae_1,&{}\left[ e_2,e_3\right] =Be_1,&{}\left[ e_1,e_4\right] =Ce_1,\\ \left[ e_2,e_4\right] =\frac{B(C-D)}{A}e_1+De_2,&{}\left[ e_3,e_4\right] =Ee_1+Fe_2,&{} \quad A\ne 0, \end{array}\)

      3.

      \(\begin{array}{lll} \left[ e_2,e_3\right] =Ae_1,&{}\left[ e_1,e_4\right] =(B+C)e_1,\\ \left[ e_2,e_4\right] =De_1+Ce_2+Ee_3,&{}\left[ e_3,e_4\right] =Fe_1+Ge_2+Be_3, \quad A\ne 0, \end{array}\)

      4.

      Open image in new window

      5.

      \(\begin{array}{ll} \left[ e_1,e_3\right] =Ae_2,&{}\left[ e_1,e_4\right] =(B-C)e_1+De_2+Ee_3,\\ \left[ e_2,e_4\right] =Be_2,&{}\left[ e_3,e_4\right] =Fe_1+Ge_2+Ce_3, \quad A\ne 0, \end{array}\)

      6.

      \(\begin{array}{ll} \left[ e_2,e_3\right] =Ae_2,&{}\left[ e_1,e_4\right] =Be_1,\\ \left[ e_2,e_4\right] =Ce_2,&{}\left[ e_3,e_4\right] =De_1+Ee_2,\quad A\ne 0, \end{array}\)

      7.

      \(\begin{array}{lll} \left[ e_1,e_3\right] =Ae_2,&{}\left[ e_2,e_3\right] =Be_2,&{}\left[ e_1,e_4\right] =\frac{AC-BD}{A}e_1+De_2,\\ \left[ e_2,e_4\right] =Ce_2,&{}\left[ e_3,e_4\right] =Ee_1+Fe_2,&{}\quad A\ne 0, \end{array}\)

      8.

      \({\begin{array}{ll} \left[ e_1,e_2\right] =Ae_3,&{}\left[ e_1,e_3\right] =Be_3,\\ \left[ e_2,e_3\right] =Ce_3,&{}\left[ e_1,e_4\right] =-\frac{CD}{B}e_1+De_2+Ee_3,\\ \left[ e_2,e_4\right] =-\frac{CF}{B}e_1+Fe_2+Ge_3,&{}\left[ e_3,e_4\right] =\frac{A(BF-CD)+B(BG-CE)}{AB}e_3, \quad AB\ne 0, \end{array}}\)

      9.

      \(\begin{array}{ll} \left[ e_1,e_2\right] =Ae_3,&{}\left[ e_1,e_4\right] =Be_1+Ce_2+De_3,\\ \left[ e_2,e_4\right] =E_1+Fe_2+Ge_3,&{}\left[ e_3,e_4\right] =(B+F)e_3, \quad A\ne 0, \end{array}\)

      10.

      \(\begin{array}{lll} \left[ e_1,e_2\right] =Ae_3,&{}\left[ e_2,e_3\right] =Be_3,&{}\left[ e_1,e_4\right] =Ce_1+De_3,\\ \left[ e_2,e_4\right] =Ee_1+Fe_3,&{}\left[ e_3,e_4\right] =\frac{AC-BD}{A}e_3,&{}\quad A\ne 0, \end{array}\)

      11.

      \({\begin{array}{lll} \left[ e_1,e_3\right] =Ae_3,&{}\left[ e_2,e_3\right] =Be_3,&{}\left[ e_1,e_4\right] =Ce_1+\frac{AC}{B}e_2+\frac{AD}{B}e_3,\\ \left[ e_2,e_4\right] =Ee_1-\frac{AE}{B}e_2+De_3,&{}\left[ e_3,e_4\right] =Fe_3,&{}\quad B\ne 0, \end{array}}\)

      12.

      \(\begin{array}{ll} \left[ e_1,e_3\right] =Ae_3,&{}\left[ e_1,e_4\right] =Be_2+Ce_3,\\ \left[ e_2,e_4\right] =De_2,&{}\left[ e_3,e_4\right] =Ee_3,\quad A\ne 0. \end{array}\)

       
    3. (a3)
      \({\mathfrak {g}}'_3\)is two dimensional: In this case \({\mathfrak {g}}=\mathfrak e(2)\rtimes \mathfrak {r}\) or \({\mathfrak {g}}=\mathfrak e(1,1)\rtimes \mathfrak {r}\) and \({\mathfrak {g}}\) is isometric to one of the following Lie algebras : 
      1.

      \({\begin{array}{lll} \left[ e_1,e_3\right] =Ae_1+Be_2,&{}\left[ e_2,e_3\right] =Ce_1+De_2,&{}\left[ e_1,e_4\right] =\frac{AE+CF-DE}{C}e_1+\frac{BE}{C}e_2,\\ \left[ e_2,e_4\right] =Ee_1+Fe_2,&{}\left[ e_3,e_4\right] =Ge_1+He_2,&{} \quad C\ne 0,\ AD-BC\ne 0, \end{array}}\)

      2.

      \({\begin{array}{lll} \left[ e_1,e_3\right] =Ae_1,&{}\left[ e_2,e_3\right] =Be_2,&{}\left[ e_1,e_4\right] =Ce_1,\\ \left[ e_2,e_4\right] =De_2,&{}\left[ e_3,e_4\right] =Ee_1+Fe_2,\quad AB\ne 0, \end{array}}\)

      3.

      \({\begin{array}{lll} \left[ e_1,e_3\right] =Ae_1,&{}\left[ e_2,e_3\right] =Ae_2,&{}\left[ e_1,e_4\right] =Be_1+C e_2,\\ \left[ e_2,e_4\right] =De_1+Ee_2,&{}\left[ e_3,e_4\right] =Fe_1+Ge_2,\quad A\ne 0, \end{array}}\)

      4.

      \({\begin{array}{lll} \left[ e_1,e_3\right] =Ae_1+Be_2,&{}\left[ e_2,e_3\right] =Ce_2,&{}\left[ e_1,e_4\right] =\frac{BD+AE-CE}{B}e_1+E e_2,\\ \left[ e_2,e_4\right] =De_2,&{}\left[ e_3,e_4\right] =Fe_1+Ge_2,&{}\quad B\ne 0,\ AC\ne 0, \end{array}}\)

      5.

      \({\begin{array}{lll} \left[ e_1,e_2\right] =Ae_1+Be_3,&{}\left[ e_2,e_3\right] =Ce_1+De_3,&{}\left[ e_1,e_4\right] =\frac{CH-AG-DG}{C}e_1-\frac{BG}{C}e_3\\ \left[ e_2,e_4\right] =Ee_1+Fe_3,&{}\left[ e_3,e_4\right] =Ge_1+He_3,&{} \quad C\ne 0,\ AD-BC\ne 0, \end{array}}\)

      6.

      \({\begin{array}{lll} \left[ e_1,e_2\right] =Ae_1,&{}\left[ e_2,e_3\right] =Be_3,&{}\left[ e_1,e_4\right] =Ce_1,\\ \left[ e_2,e_4\right] =De_1+Ee_3,&{}\left[ e_3,e_4\right] =Fe_3,\quad AB\ne 0, \end{array}}\)

      7.

      \({\begin{array}{lll} \left[ e_1,e_2\right] =Ae_1, &{}\left[ e_2,e_3\right] =-Ae_3,&{}\left[ e_1,e_4\right] =Be_1+Ce_3,\\ \left[ e_2,e_4\right] =De_1+Ee_3,&{}\left[ e_3,e_4\right] =Fe_1+Ge_3,\quad A\ne 0, \end{array}}\)

      8.

      \({\begin{array}{lll} \left[ e_1,e_2\right] =Ae_1+Be_3, &{}\left[ e_2,e_3\right] =Ce_3,&{}\left[ e_1,e_4\right] =\frac{CD+AD+BE}{B}e_1+De_3,\\ \left[ e_2,e_4\right] =Fe_1+Ge_3,&{}\left[ e_3,e_4\right] =Ee_3,&{}\quad B\ne 0,\ AC\ne 0, \end{array}}\)

      9.

      \({\begin{array}{lll} \left[ e_1,e_2\right] =Ae_2,&{}\left[ e_1,e_3\right] =Be_3,&{}\left[ e_1,e_4\right] =Ce_2+De_3,\\ \left[ e_2,e_4\right] =Ee_2,&{}\left[ e_3,e_4\right] =Fe_3,\quad AB\ne 0, \end{array}}\)

      10.

      \({\begin{array}{lll} \left[ e_1,e_2\right] =Ae_2,&{}\left[ e_1,e_3\right] =Ae_3,&{}\left[ e_1,e_4\right] =Be_2+Ce_3,\\ \left[ e_2,e_4\right] =De_2+Ee_3,&{}\left[ e_3,e_4\right] =Fe_2+Ge_3,\quad A\ne 0, \end{array}}\)

      11.

      \({\begin{array}{lll} \left[ e_1,e_2\right] =Ae_2+Be_3,&{}\left[ e_1,e_3\right] =Ce_3,&{}\left[ e_1,e_4\right] =Ce_2+Ee_3,\\ \left[ e_2,e_4\right] =\frac{AF+BG-CF}{B}e_2+Fe_3,&{}\left[ e_3,e_4\right] =Ge_3,&{} \quad B\ne 0,\ AC\ne 0, \end{array}}\)

      12.

      \({\begin{array}{lll} \left[ e_1,e_2\right] =Ae_2+Be_3,&{}\left[ e_1,e_3\right] =Ce_2+De_3,&{}\left[ e_1,e_4\right] =Ee_2+Fe_3,\\ \left[ e_2,e_4\right] =\frac{AG+CH-DG}{C}e_2+\frac{BG}{C}e_3,&{}\left[ e_3,e_4\right] =Ge_2+He_3,&{} \quad C\ne 0,\ AD-BC\ne 0. \end{array}}\)

       
     
  2. (b)
    There exist a basis \(\{e_1,\dots ,e_4\}\) for \({\mathfrak {g}}\) such that \({\mathfrak {g}}_3=\mathrm{Span}(e_1,e_2,e_3)\) and \(\mathfrak {r}=\mathrm{Span}(e_4)\) and the non-zero inner products is determined by \(g(e_1,e_1)=-g(e_2,e_2)=g(e_3,e_4)=g(e_4,e_3)=1\) and \({\mathfrak {g}}\) is isometric to one of the following Lie algebras : 
    1.

    \({\begin{array}{lll} \left[ e_1,e_2\right] =-e_3, &{}\left[ e_1,e_3\right] =-Ae_1,&{}\left[ e_2,e_3\right] =Ae_2,\\ \left[ e_1,e_4\right] =-Be_1+Ce_3,&{}\left[ e_2,e_4\right] =Be_2+De_3,&{}\left[ e_3,e_4\right] =ADe_1+ACe_2,\quad A\ne 0, \end{array}}\)

    2.

    \({\begin{array}{lll} \left[ e_1,e_2\right] =Ae_3, &{}\left[ e_1,e_3\right] =-Ae_2,&{}\left[ e_2,e_3\right] =Ae_1,\\ \left[ e_1,e_4\right] =-B e_2-Ce_3,&{}\left[ e_2,e_4\right] =Be_1-De_3,&{}\left[ e_3,e_4\right] =Ce_1+De_2,\quad A\ne 0. \end{array}}\)

     

Remark 2.4

In the classifications presented in the Theorems 2.2 and 2.3, the cases are generally non-equivalent but obviously one can find the same Lie algebras from different cases, by assigning suitable values to the arbitrary coefficients. For example, the case (a1) in the Theorem 2.2 for \(A+C+D=0\), is the same Lie algebra of the case (a2) with \(\beta =0\).

3 Geometry of Four-Dimensional Lie Groups

Based on the classification which presented in the Sect. 2, we study geometry of each class in this section. It is well known that geometry of different spaces is related to their Riemann curvature tensor, the (1, 3)-tensor field which is defined by the identity \(R(X,Y)Z=[\Lambda (X),\Lambda (Y)]Z-\Lambda ([X,Y])Z\), where XYZ are arbitrary tangent vector fields on the base manifold and \(\Lambda \) is the Levi-Civita connection deduced by the well known Koszul formula,
$$\begin{aligned} 2g(\Lambda (X)Y,Z)=g(X,[Z,Y])+g(Y,[Z,X])+g(Z,[X,Y]). \end{aligned}$$
Finally, the Ricci tensor is defined by the following contraction on the curvature tensor’s indices,
$$\begin{aligned} \varrho (X,Y)=\text {Tr}\{Z\rightarrow R(Z,X)Y\}. \end{aligned}$$
Several geometric properties are related to the Ricci tensor. A manifold (Mg) is called Ricci flat if it’s Ricci tensor vanishes identically. A more general condition appears in Einstein property, that is \(\varrho =\lambda g\), for a real constant \(\lambda \). Ricci parallel condition is also means that the covariant derivative of the Ricci tensor will be zero. Clearly, each Ricci flat manifold is Einstein and every Einstein manifold is Ricci parallel.
If the covariant derivative of the curvature tensor vanishes, i.e., \(\Lambda (R)=0\), the manifold is called locally symmetric. Also, (Mg) is called conformally flat if the Weyl conformal tensor vanishes identically, where W is defined by,
$$\begin{aligned} W(X,Y,Z,V)= & {} R(X,Y,Z,V)+\frac{\tau }{6}\{g(X,V)g(Y,Z)-g(X,Z)g(Y,V)\}\nonumber \\&-\,\frac{1}{2}\{g(X,V)\varrho (Y,Z)-\varrho (X,Z)g(Y,V)+ \varrho (X,V)g(Y,Z)\nonumber \\&-\,g(X,Z)\varrho (Y,V)\}, \end{aligned}$$
(3.1)
for arbitrary tangent vector fields \(X,Y,Z,V\in \mathfrak X(M)\).

Theorem 3.1

Let (Gg) be one of the Lie groups listed in the Theorems 2.2 and 2.3, Einstein and Ricci parallel examples are listed in the following Tables 1 and 2.

Table 1

Einstein and Ricci parallel examples of the Theorem 2.2

Case

Ricci flat

Einstein not Ricci flat

Ricci parallel not Einstein

a1

\(A=C+D=0, B=2\varepsilon \alpha \)

\((2A-D)^2=C^2=(D^2+8\alpha ^2),B=0\)

\(B=0,CA=-DA=\alpha ^2\)

a2

\(A+B=C+2\varepsilon \alpha =\beta =0\)

b1

\(A=B=CD+\gamma ^2=0\)

b2

\(A=B=C+\varepsilon \gamma =D+\varepsilon \beta =0\)

b3

c1

\(B=C=\alpha -\beta =\gamma =0\)

\(\alpha =\gamma =\beta \)

c2

\(A=C=D=\gamma -\alpha =0\)

\(B=C=D=\alpha +\gamma =0,A=\varepsilon \gamma \)

\(\begin{array}{l} A=D=\gamma =B^2-C^2-\alpha ^2=0\\ C=D=\alpha -\gamma =0\\ C=D=\alpha +\gamma =A^2+B^2-\alpha ^2=0\\ A=C=D=\beta +\varepsilon \alpha =0\end{array}\)

c3

\(A=B=D=\gamma -\beta =0\)

\(A=B=C=\gamma +\beta =D+\varepsilon \beta =0\)

\(\begin{array}{l} B=D=A^2-C^2+\beta ^2=\gamma =0\\ A=B=D=C+\varepsilon \beta =0\\ A=B=\beta -\gamma =0\\ A=B=D^2+C^2-\beta ^2=\gamma +\beta =0\end{array}\)

c4

\(A=C=D=\beta -\alpha =0\)

\(B=C=D=\alpha +\beta =A+\varepsilon \beta =0\)

\(\begin{array}{l}A=D=B^2-C^2+\alpha ^2=\beta =0\\ C=D=\alpha -\beta =0\\ C=D=A^2-B^2-\alpha ^2=\beta +\alpha =0\end{array}\)

c5

\(\begin{array}{l}A+\varepsilon \gamma =2B+\varepsilon \gamma =C+E\\ \quad =D=F=0\end{array}\)

\(A=B=C=E=D^2-F^2+\gamma ^2=0\)

c6

\(\begin{array}{l}A-\varepsilon \beta =2B-\varepsilon \beta \\ \quad =C=D-E=F=0,\ or\\ A-\delta \beta =2B+\delta \beta +2\varepsilon E\\ \quad =C=D+E+2\delta \varepsilon \beta =F=0\\ \end{array}\)

\(\begin{array}{l}A=B=C^2-F^2+\beta ^2=D=E=0\end{array}\)

c7

\(\begin{array}{l} B-A=C=E=F-D\\ \quad =\alpha -2\varepsilon A=0,\ or\\ C=2D-\varepsilon (B+3A)=E\\ \quad =2F-\varepsilon (A+3B)=\alpha -\delta (A+B)=0\end{array}\)

\(A=B=C^2-E^2+\alpha ^2=D=F=0\)

c8

\(\begin{array}{l}A+K+E=C+G=F=H=0,\\ B=D=\varepsilon \sqrt{G^2-E^2-KE-K^2}\end{array}\)

\(\begin{array}{l} A-K=B=C-G\\ \quad =D=E-K=F=H=0,\ or\\ A+2K=B=C=D\\ \quad =E+5K=F+6K=G=H=0\end{array}\)

\(\begin{array}{l} AK+C^2=B=D=E\\ \quad =F=G+C=H=0\end{array}\)

d1

\(\begin{array}{l}A=B=C+D=0,\beta =\varepsilon \ or\ \\ A=B=0,C=-D=\pm 1\end{array}\)

\(A+\varepsilon B=C+D=0,\beta =\varepsilon \)

d2

\(\begin{array}{l}A=B+\delta D=C-\varepsilon \delta D\\ \quad =E+F=0\end{array}\)

\(B+D=C-\varepsilon D=E-A=F-A=0\)

d3

d4

\(A+B=C-\delta D=\alpha =0\)

\(A-B+2\delta =C=D=\alpha -\varepsilon (2\delta B-2)=0\)

\(A+B=\alpha =0\)

e1

\(A=B=C=D=\beta +\gamma =\alpha -\delta =0\)

e2

\(C=D=B-\varepsilon \alpha =A+\varepsilon \delta =0\)

\(\begin{array}{l} AB+\alpha \delta =C=D=0,\ or\\ A-B=C=D=\alpha -\delta =0\end{array}\)

e3

\(A+D=B-C=E=F=\alpha ^2-C^2-D^2=0\)

\(\begin{array}{l}A=B+C=D=E=F=0,\ or\\ B-C=AD-C^2+\alpha ^2=E=F=0,\ or\\ A-D=B^2-D^2-\alpha ^2\\ \quad =C^2-D^2-\alpha ^2=E=F=0\end{array}\)

e4

\(A=B+\varepsilon \alpha =C=D=\beta =0\)

\(\begin{array}{l} A=C=D=\beta =0,\ or\\ B=C=D=\beta =0\end{array}\)

f1

\(\begin{array}{l}A=B=C-\varepsilon \gamma =0,\\ D=\delta =\frac{\gamma (\beta -\gamma )}{\pm \sqrt{2(\beta ^2+\gamma ^2)}}\end{array}\)

\(\begin{array}{l}A=B=\alpha -\delta =\beta -\gamma =0,\ or \\ A=B=C-\varepsilon \gamma =D-\varepsilon \delta =0,\ or\\ A-\varepsilon B=C-\varepsilon D=\alpha -\varepsilon \beta = \delta -\varepsilon \gamma =0\end{array}\)

f2

\(\begin{array}{l} A=B=CD-\alpha \delta =0, \ or\\ A=B=C-D=\alpha -\delta =0\end{array}\)

f3

\(A=B=C+F=0,D=-E=\pm \sqrt{F^2+\alpha ^2}\)

\(\begin{array}{l} A=B=CF+E^2-\alpha ^2=D+E=0,\ or\\ A=B=C-F=D-E=0\end{array}\)

f4

\(\begin{array}{l} A=B=C-\varepsilon \alpha \\ \quad =D=\beta ^2-2\alpha ^2=0\end{array}\)

\(\begin{array}{l} A=B=CD=\beta =0,\ or\\ A=B=C-\varepsilon \alpha =D=0,\ or\\ A-\varepsilon B=D=\alpha -\varepsilon \beta =0\end{array}\)

g1

\(A-\varepsilon \gamma =B+C-\varepsilon \gamma =D=2(\beta +\varepsilon C)- \gamma =0\)

g2

\(A=B+C=D^2-2\alpha ^2=0\)

\(\begin{array}{l} 2A-C-B=4\alpha ^2-B^2+C^2\\ \quad =2D\alpha +\beta (B+C)=0\end{array}\)

\(\begin{array}{l}A=B+C=0\ or\\ B\alpha ^2 =-C\alpha ^2=A\beta ^2,D\alpha +A\beta =0\end{array}\)

Table 2

Einstein and Ricci parallel examples of the Theorem 2.3

Case

Ricci flat

Einstein not Ricci flat

Ricci parallel not Einstein

a1

\(\begin{array}{l} G=H=2K(A+E)\\ \quad +\,(D-B)^2-2(A^2+E^2)=0\end{array}\)

\(\begin{array}{l}G=H=K=0,\ or\\ A+\varepsilon B=CG-AK+\varepsilon KD-\varepsilon GF\\ \quad =E+\varepsilon D=H+\varepsilon G=0\end{array}\)

a2-1

\(\begin{array}{l} 2B^2-C^2=2C^2D^2-A^4\\ \quad =EC+A^2=F=G=0\end{array}\)

\(\begin{array}{l} B=C=E=AF-DG=0,\ or\\ C=E=F=G=0,\ or\\ 2BD-A^2=2EB^2+CA^2=F=G=0\end{array}\)

a2-2

\(B^2-2A^2=C=D=E=F=0\)

\(BC=D=E=F=0\)

a2-3

\(\begin{array}{l}2AD+F^2=B=C=E=G=0 \end{array}\)

a2-4

\(\begin{array}{l} B^2-2C^2=2C^2D^2-A^4\\ \quad =E=2CF+A^2=G=0\end{array}\)

\(\begin{array}{l} B=D=E=G=0,\ or\\ B=C=D=AE-FG=0,\ or\\ 2DC^2-BA^2=E=2CF+A^2=G=0\end{array}\)

a2-5

\(\begin{array}{l}2AD-G^2=B=C=E=F=0\end{array}\)

a2-6

\(\begin{array}{l}B=D=E=0 \end{array}\)

a2-7

\(A^2-2B^2=C=D=E=F=0\)

\(\begin{array}{l}B=C=2AD-F^2=E=0,\ or\\ C=D=E=F=0\end{array}\)

a2-8

\(\begin{array}{l} C+\varepsilon B=AD+EB-A^2\\ \quad +\,\varepsilon (AF+BG)=0,\ or\\ D=F=0,\ or\\ C+\varepsilon B=D+\varepsilon F=0 \end{array}\)

a2-9

\(4BF-A^2+(C-E)^2=0\)

\(B+F=0\)

a2-10

\(A+\varepsilon E=B=0\)

\(C=BE=0\)

a2-11

\(\begin{array}{l} {A+\varepsilon B=F=0},\ or\\ C=E=0,\ or\\ A+\varepsilon B=C+\varepsilon E=0\end{array}\)

a2-12

\(B=D=0\)

a3-1

\(\begin{array}{l} 2A^2-(C-B)^2+2D^2\\ \quad =E=F=G=H=0,\ or\\ A=B-C=D=\\ F=G=H=0\end{array}\)

\(\begin{array}{l} { A+\varepsilon \sqrt{2}C}=B+C\\ \quad =D={\varepsilon \sqrt{2}E-F}=G=H=0,\ or\\ A^2-{2C^2+D^2}=B+C\\ \quad =FC+AE=G=H=0,\ or\\ A={\varepsilon \sqrt{2}B+D=\varepsilon \sqrt{2}C-D}\\ \quad =F=G=H=0\end{array}\)

\(\begin{array}{l}E=F=G=H=0,\ or\\ B+\varepsilon C=A+\varepsilon D\\ \quad =E=G=H=0,\ or\\ B+C=FC(A+D)\\ \qquad +\,E(AD-D^2+2C^2)=G=H=0 \end{array}\)

a3-2

\(\begin{array}{l}{BC+AD=E=F=0},\ or\\ A-B=C-D=E=F=0,\ or\\ 2A-B=D+2C=E=F=0\end{array}\)

a3-3

\(B+\varepsilon E=C+\varepsilon D=F=G=0\)

a3-4

\(\begin{array}{l} 2A^2-B^2+2C^2=D\\ \quad =E=F=G=0\end{array}\)

\(D=E=F=G=0\)

a3-5

a3-6

\(AE+C(C-F)=B+2A=D=0\)

a3-7

\(\begin{array}{l} A+\varepsilon F=D+\varepsilon B\\ \quad =2EF+\varepsilon (2BG-B^2-2CF)=0\end{array}\)

a3-8

\(\begin{array}{l} C+2A=B-F\\ \quad =GB^2+D(AD-EB)=0\end{array}\)

\(\begin{array}{l} C=F+B\\ \quad =GAB-DAE-BE^2+B^3=0\end{array}\)

a3-9

\(AD+E(E-F)=B-2A=C=0\)

a3-10

\(\begin{array}{l} A+\varepsilon F=B+\varepsilon D\\ \quad =2CF-\varepsilon (-2DG+D^2+2EF)=0\end{array}\)

a3-11

\(\begin{array}{l} 4AE+F(F-2G)=B-2A\\ \quad =C-2A=0\end{array}\)

a3-12

b1

b2

In the above tables, ✗ in each block means that the corresponding condition never establishes for that case and we always have \(\varepsilon ^2=1\).

Proof

The proof is based on case by case study of the Theorems 2.2 and 2.3. Starting from the case (a1) of the Theorem 2.2, we bring computations in details and the other cases will be treated in the similar way.

We set \(\Lambda [i]=\Lambda (e_i)\) for all indices \(i=1,\dots ,4\) and by applying the well known Koszul formula, the components of the Levi-Civita connection are deduced by standard calculations.
$$\begin{aligned}&{}\Lambda [1]=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0 &{} \alpha &{}-\alpha &{}A \\ -\alpha &{}0 &{}0 &{}\dfrac{B}{2}\\ -\alpha &{}0 &{}0 &{}\dfrac{B}{2} \\ A &{}\dfrac{B}{2} &{}-\dfrac{B}{2} &{}0 \end{array}\right) ,\quad \Lambda [2]=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0 &{} 0 &{}0 &{}\dfrac{B}{2} \\ 0 &{}0 &{}\alpha &{}C\\ 0 &{}\alpha &{}0 &{}\dfrac{C-D}{2} \\ \dfrac{B}{2} &{} C&{}\dfrac{D-C}{2} &{}0 \end{array}\right) ,\nonumber \\&{}\Lambda [3]=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0 &{} 0 &{}0 &{}-\dfrac{B}{2} \\ 0 &{}0 &{}-\alpha &{}\dfrac{D-C}{2}\\ 0 &{}-\alpha &{}0 &{}D \\ -\dfrac{B}{2} &{}\dfrac{D-C}{2} &{}-D &{}0 \end{array}\right) ,\quad \Lambda [4]= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0&{}-\dfrac{B}{2}&{}\dfrac{B}{2}&{}0\\ \dfrac{B}{2}&{}0&{}-\dfrac{C+D}{2}&{}0\\ \dfrac{B}{2}&{}-\dfrac{C+D}{2}&{}0&{}0\\ 0&{}0&{}0&{}0\end{array} \right) ,\nonumber \\ \end{aligned}$$
(3.2)
where these matrices are obtained with respect to the pseudo-orthonormal basis \(\{e_i\}_{i=1}^4\) with \(e_3\) and \(e_4\) time-like.
The curvature tensor will be determined by the following non-zero components (by setting \(R_{ij}=R(e_i,e_j)\), \(i,j=1,\dots ,4\)).
$$\begin{aligned} R_{12}= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0 &{}-2\alpha ^2+AC-\dfrac{B^2}{4} &{}2\alpha ^2-\dfrac{A(C-D)}{2}+\dfrac{B^2}{4} &{}\dfrac{\alpha (C+D)}{2}-\alpha A \\ 2\alpha ^2-AC+\dfrac{B^2}{4} &{}0 &{}\dfrac{B(C+D)}{4} &{}-\dfrac{3\alpha B}{2}\\ 2\alpha ^2-\dfrac{A(C-D)}{2}+\dfrac{B^2}{4} &{}\dfrac{B(C+D)}{4} &{}0 &{} -\dfrac{3\alpha B}{2}\\ \dfrac{\alpha (C+D)}{2}-\alpha A&{}-\dfrac{3\alpha B}{2} &{}\dfrac{3\alpha B}{2} &{}0 \end{array}\right) ,\\ R_{13}= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0 &{}2\alpha ^2-\dfrac{2A(C-D)-B^2}{4} &{}-2\alpha ^2-AD-\dfrac{B^2}{4} &{}-\dfrac{\alpha (C+D)}{2}+\alpha A \\ -2\alpha ^2+\dfrac{2A(C-D)-B^2}{4} &{}0 &{}-\dfrac{B(C+D)}{4} &{}\dfrac{3\alpha B}{2}\\ -2\alpha ^2-AD-\dfrac{B^2}{4} &{}-\dfrac{B(C+D)}{4} &{}0 &{}\dfrac{3\alpha B}{2} \\ -\dfrac{\alpha (C+D)}{2}+\alpha A&{}\dfrac{3\alpha B}{2} &{} -\dfrac{3\alpha B}{2}&{}0 \end{array}\right) ,\\ R_{14}= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0 &{} \dfrac{\alpha (C+D)}{2}-\alpha A &{}-\dfrac{\alpha (C+D)}{2}+\alpha A &{}-A^2 \\ -\dfrac{\alpha (C+D)}{2}+\alpha A &{}0 &{}0 &{}-AB+\dfrac{B(C+D)}{4}\\ -\dfrac{\alpha (C+D)}{2}+\alpha A &{} 0&{}0 &{}-AB+\dfrac{B(C+D)}{4} \\ -A^2&{}-AB+\dfrac{B(C+D)}{4} &{}AB-\dfrac{B(C+D)}{4} &{}0 \end{array}\right) ,\\ R_{23}= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0&{}\dfrac{B(C+D)}{4} &{}-\dfrac{B(C+D)}{4} &{}0 \\ -\dfrac{B(C+D)}{4} &{}0 &{}-\dfrac{(C+D)^2}{4} &{}0\\ -\dfrac{B(C+D)}{4} &{}-\dfrac{(C+D)^2}{4} &{}0 &{}0\\ 0&{} 0&{}0 &{}0 \end{array}\right) ,\\ R_{24}= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0 &{}- \dfrac{3\alpha B}{2} &{}\dfrac{3\alpha B}{2} &{}-AB+\dfrac{B(C+D)}{4} \\ \dfrac{3\alpha B}{2} &{}0 &{}0 &{}-\dfrac{3B^2+(C+D)^2}{4}\\ \dfrac{3\alpha B}{2}&{} 0&{}0 &{}-\dfrac{3B^2}{4} \\ -AB+\dfrac{B(C+D)}{4}&{}-\dfrac{3B^2+(C+D)^2}{4} &{}\dfrac{3B^2}{4} &{}0 \end{array}\right) , \end{aligned}$$
and finally
$$\begin{aligned} R_{34}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0&{}\dfrac{3\alpha B}{2} &{} -\dfrac{3\alpha B}{2}&{} AB-\dfrac{B(C+D)}{4}\\ -\dfrac{3\alpha B}{2} &{}0 &{}0 &{}\dfrac{3B^2}{4}\\ -\dfrac{3\alpha B}{2}&{}0 &{}0 &{} -\dfrac{-3B^2+(C+D)^2}{4}\\ AB-\dfrac{B(C+D)}{4}&{}\dfrac{3B^2}{4} &{}\dfrac{-3B^2+(C+D)^2}{4} &{}0 \end{array}\right) .\nonumber \\ \end{aligned}$$
(3.3)
Consequently, the Ricci tensor with respect to the pseudo-orthonormal basis \(\{e_i\}_{i=1}^4\) is given by
Table 3

Locally symmetric and conformally flat examples of the Theorem 2.2

Case

Locally symmetric

Conformally flat

a1

\(\begin{array}{l} AC=-AD=\alpha ^2,B=0\ or\\ AC-A^2=A^2-AD=2\alpha ^2,B=0\end{array}\)

\( AC-A^2=A^2-AD=2\alpha ^2,B=0\)

a2

b1

\(A=B=CD+\gamma ^2=0\)

\(A=B=0,C=-D=\pm \gamma \)

b2

b3

c1

\(\begin{array}{l} B=C=\alpha -\beta =\gamma =0\ or\\ \alpha =\beta =\gamma \end{array}\)

\(\begin{array}{l} B=C=\alpha -\beta =\gamma =0\ or\\ \alpha =\beta =\gamma \end{array}\)

c2

\(\begin{array}{l} C=D=\alpha -\gamma =0\ or\\ C=D=\alpha +\gamma =A^2+B^2-\alpha ^2=0\end{array}\)

\(\begin{array}{l} C=D=\alpha -\gamma =0,\ or \\ A=C=D=\alpha +\gamma =B+\varepsilon \gamma =0 \end{array}\)

c3

\(\begin{array}{l} A=B=\beta -\gamma =0\ or\\ A=B=C^2+D^2-\beta ^2=\beta +\gamma =0\end{array}\)

\(\begin{array}{l} A=B=\beta -\gamma =0\ or \\ A=B=D=\beta +\gamma =C+\varepsilon \gamma =0 \end{array}\)

c4

\(\begin{array}{l} C=D=\alpha -\beta =0,\ or\\ A^2-B^2-\alpha ^2=C=D=\beta +\alpha =0 \end{array}\)

\(C=D=\alpha -\beta =0\)

c5

\(A+\varepsilon \gamma =2B+\varepsilon \gamma =C+E=D=F=0 \)

c6

\(A+\varepsilon \beta =2B+\varepsilon \beta =C=D-E=F=0\)

c7

\(A-B=C=D-F=E=\alpha +2\varepsilon B=0\)

c8

\(\begin{array}{l} AK+C^2=B=D=E\\ \quad =F=C+G=H=0\end{array}\)

\(\begin{array}{l} A+D=B-D=C=E+D\\ \quad =F=G=H=K+2D=0\end{array}\)

d1

\(A+\varepsilon B=C+D=\beta +\varepsilon =0\)

\(\begin{array}{l} B+\varepsilon A=C+D=\beta -\varepsilon =0,\ or\\ A=B=\beta -\varepsilon =0,C=-D=\pm 1\ or\\ A=B=C-D+2\delta =2\beta -\varepsilon (\delta D+1)=0 \end{array}\)

d2

\(\begin{array}{l} A=B+\varepsilon C=D=E+F=0,\ or\\ A=B+D=C-\varepsilon D=E+F=0,\ or\\ A-F=B+D=C-\varepsilon D=E-F=0 \end{array}\)

\(\begin{array}{l} A=B=C=D=0,E=-F=\pm 1,\ or\\ A=B+\varepsilon C=D=E+F=0,\ or\\ B+D=C-\varepsilon D=E-2A+F=0 \end{array}\)

d3

d4

\(\begin{array}{l} A+B=D=\alpha =0,\ or\\ A+B=C-\varepsilon D=\alpha =0,\ or\\ A-B+2\delta =C=D=\alpha -\varepsilon (2\delta B-2)=0 \end{array}\)

\(\begin{array}{l} A+B=D=\alpha =0,\ or\\ A+B=C-\varepsilon D=\alpha =0,\ or\\ C=D=\alpha =0,A=-B=\pm 1 \end{array}\)

e1

\(A=B=C=D=\beta +\gamma =\alpha -\delta =0\)

\(C=D=\beta +\gamma =\alpha -\delta =0\)

e2

\(\begin{array}{l} AB+\alpha \delta =C=D=0,\ or\\ A-B=C=D=\alpha -\delta =0 \end{array}\)

\(A-B=C=D=\alpha -\delta =0\)

e3

\(\begin{array}{l} A=B+C=D=E=F=0,\ or\\ B-C=AD-C^2+\alpha ^2=E=F=0,\ or\\ A-D=B^2-D^2-\alpha ^2=C^2-D^2-\alpha ^2=E=F=0\end{array}\)

\(A-D=B+C=E=F=0\)

e4

\(\begin{array}{l} A=C=D=\beta =0,\ or\\ B=C=D=\beta =0 \end{array}\)

f1

\(\begin{array}{l} A=B=\alpha -\delta =\beta -\gamma =0,\ or\\ A=B=C+\varepsilon D=\alpha +\varepsilon \beta =\delta +\varepsilon \gamma =0,\ or\\ A+\varepsilon B=C=D=\alpha +\varepsilon \beta =\delta +\varepsilon \gamma =0,\ or\\ A+\varepsilon B=C+\varepsilon D=\alpha =\beta =\delta +\varepsilon \gamma =0,\ or\\ A+\varepsilon B=C-\gamma =D+\varepsilon \gamma =\alpha +\varepsilon \beta =\delta +\varepsilon \gamma =0,\ or\\ A+\varepsilon B=C+\gamma =D-\varepsilon \gamma =\alpha +\varepsilon \gamma =\beta -\gamma =\delta +\varepsilon \gamma =0,\ or\\ A+\varepsilon B=C+\gamma =D-\varepsilon \gamma =\alpha -2\varepsilon \gamma =\beta +2\gamma =\delta +\varepsilon \gamma =0\ or\\ A+\varepsilon B=C+\gamma =D-\varepsilon \gamma =\alpha +2\varepsilon \gamma =\beta -2\gamma =\delta +\varepsilon \gamma =0 \end{array}\)

\(\begin{array}{l} A=B=\alpha -\delta =\beta -\gamma =0,\ or\\ A+\varepsilon B=C+\varepsilon D=\alpha +\varepsilon \beta =\delta +\varepsilon \gamma =0\end{array}\)

f2

\(\begin{array}{l}A=B=CD-\alpha \delta =0, \ or\\ A=B=C-D=\alpha -\delta =0\end{array}\)

\(\begin{array}{l} A=B=\alpha -\delta =0,\ or\\ A=B=\alpha +\varepsilon D=\delta +\varepsilon C=0,\ or\\ A=B=C+2\varepsilon \alpha =D+\varepsilon \alpha =\delta -2\alpha =0,\ or\\ A=B=C+{\varepsilon \delta =D}+2\varepsilon \delta =\alpha -2\delta =0 \end{array}\)

f3

\(\begin{array}{l} A=B=CF+E^2-\alpha ^2=D+E=0,\ or\\ A=B=C-F=D-E=0,\ or\\ A=B=C-F=0,D=-E=\pm \sqrt{\alpha ^2-F^2} \end{array}\)

\(\begin{array}{l}A+\varepsilon B=C+\varepsilon D=E+\varepsilon F=\alpha =0,\ or\\ A=B=C-F=D-E=0\end{array}\)

f4

\(\begin{array}{l}A=B=CD=\beta =0,\ or\\ A+\varepsilon B=D=\alpha +\varepsilon \beta =0\end{array}\)

\(\begin{array}{l} A=B=C=D+\varepsilon \alpha =\beta =0,\ or\\ A+\varepsilon B=D=\alpha +\varepsilon \beta =0 \end{array}\)

g1

\(\begin{array}{l} A+\varepsilon \gamma =B+C+\varepsilon \gamma =2\beta -2\varepsilon C-\gamma =D=0,\ or\\ A+\varepsilon \gamma =2B+\varepsilon (2\beta +\gamma )=2C-\varepsilon (2\beta -\gamma )=D=0\end{array}\)

g2

\(\begin{array}{l}AB=-AC=D^2, A\beta +D\alpha =0,\ or\\ AB-A^2=A^2-AC=\alpha ^2 \end{array}\)

\( AB-A^2=A^2-AC=\alpha ^2\)

Table 4

Locally symmetric and conformally flat examples of the Theorem 2.3

Case

Locally symmetric

Conformally flat

a1

\(\begin{array}{l}B-D=A-E=G=H=K=0 \end{array}\)

\(\begin{array}{l} {B-D=E-A=G=H=0} \end{array}\)

a2-1

\(\begin{array}{l} B=C=E=FA-DG=0,\ or\\ C=E=F=G=0 \end{array}\)

a2-2

\(B=D=E=F=0\)

a2-3

a2-4

\(\begin{array}{l}B=C=D=EA-FG=0,\ or\\ B=D=E=G=0 \end{array}\)

a2-5

a2-6

\(B=D=E=0\)

a2-7

a2-8

\(\begin{array}{l}B+C=EB+AD-A^2\\ \qquad +\,\varepsilon (BG+AF)=0,\ or\\ D=F=0\end{array}\)

\(\begin{array}{l} C+\varepsilon B=D-\varepsilon F=0,\ or\\ D=F=0\end{array}\)

a2-9

\(\begin{array}{l}B=F=C-E=0,\ or\\ A-E-C=B+F=0,\ or\\ A-E-C=BF-EC=0,\ or\\ 2B+\varepsilon A=2F+\varepsilon A=C-E=0,\ or\\ A+E+C=B+\varepsilon E=F+\varepsilon C=0,\ or\\ 3A-E-C=3\varepsilon B-2E+C\\ \qquad =3\varepsilon F-2C+E=0\\ \end{array}\)

\(\begin{array}{l}{A-E-C=0},\ or\\ {E-C=F-B=0}\end{array}\)

a2-10

\(\begin{array}{l}{A+E=B=C+\varepsilon E=0}\ or\\ A-E=B=0,\ or\\ {C=E=0},\end{array}\)

\(\begin{array}{l}C=E=0,\ or\\ A-E=B=0\end{array}\)

a2-11

\(\begin{array}{l}C=E=0,\ or\\ A+\varepsilon B=F=0\end{array}\)

\(\begin{array}{l}{A+\varepsilon B=C-\varepsilon E=0},\ or\\ {C=E=0},\ or\\ {A+\varepsilon B=F=0} \end{array}\)

a2-12

\(\begin{array}{l}B=D=0 \end{array}\)

\(\begin{array}{l}{B=D=0},\end{array}\)

a3-1

\(\begin{array}{l}A-D=B-C=G=H=0,\ or\\ FC(A+D)+E(AD+2C^2-D^2)\\ \quad =B+C=G=H=0\end{array}\)

\(\begin{array}{l} B+C=A+D=E=G=H=0,\ or\\ B-C=A-D=G=H=0 \end{array} \)

a3-2

\(\begin{array}{l}AD+BC=E=F=0,\ or\\ A-B=C-D=E=F=0 \end{array}\)

\(\begin{array}{l}A-B=C+\varepsilon D=E=F=0,\ or\\ A+B=C-D=F=E=0 \end{array}\)

a3-3

\(\begin{array}{l}B+\varepsilon {E=C+\varepsilon D=F=G=0}, \ or\\ \end{array}\)

\(\begin{array}{l}{B+\varepsilon E=D+\varepsilon C=F=G=0},\ or\\ B+\varepsilon C=D+\varepsilon E=F=G=0 \end{array}\)

a3-4

a3-5

a3-6

\( B+2A=D=EA+C(C-F)=0\)

\(\begin{array}{l}{B+2A=D=EA+C(C-F)=0},\ or\\ B+2A=2C-F=AE+D^2-C^2=0 \end{array}\)

a3-7

\(\begin{array}{l} A+\varepsilon F=B+\varepsilon D\\ \quad ={2CF-2\varepsilon (DG-EF)}+D^2=0\end{array}\)

a3-8

\(\begin{array}{l} C+2A=F-B\\ \quad =GB^2+D(AD-BE)=0,\ or\\ C=F+B=A(BG-DE)\\ \qquad -\,B(E^2-B^2)=0\end{array}\)

\(\begin{array}{l}2A+C=BE-2AD=GAB^2\\ \qquad -\,A^2D^2+F^2B^2-FB^3=0 \end{array}\)

a3-9

\({B-2A=C=AD+E(E-F)=0}\)

\(\begin{array}{l} {B-2A=C=AD+E(E-F)=0},\ or\\ B-2A={AD+C^2-E^2}=F-2E=0 \end{array}\)

a3-10

\(\begin{array}{l} A+\varepsilon F=2CF-2BG-\varepsilon (B^2+2EF)\\ \quad =D+\varepsilon B=0 \end{array}\)

a3-11

\( {2A-C=B-C=2EC+F(F-2G)=0}\)

\(\begin{array}{l} 2A-B=C-B=2FG-2EB-F^2=0,\ or\\ 2A-C=2EB^2-CF^2+\\ 4CB^2-4B^3=BG-CF=0 \end{array}\)

a3-12

b1

\(B=C=D=k=0\)

b2

\(B=C=D=0\)

$$\begin{aligned} \varrho _{ij}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} A(C+D)+A^2 &{}AB &{} -AB&{} 0\\ AB &{}-2\alpha ^2+AC+\frac{B^2+(C+D)^2}{2} &{}2\alpha ^2-\dfrac{A(C-D)+B^2}{2} &{}\dfrac{\alpha (C+D)}{2}-\alpha A \\ -AB &{}2\alpha ^2-\dfrac{A(C-D)+B^2}{2} &{}-2\alpha ^2-AD+\frac{B^2-(C+D)^2}{2} &{}-\dfrac{\alpha (C+D)}{2}+\alpha A \\ 0 &{}\dfrac{\alpha (C+D)}{2}-\alpha A &{}-\dfrac{\alpha (C+D)}{2}+\alpha A&{} -A^2-\dfrac{(C+D)^2}{2}\\ \end{array}\right) .\nonumber \\ \end{aligned}$$
(3.4)
So the corresponding space is Ricci flat (i.e., \(\varrho _{ij}=0,\ i=1,\dots ,4\)) if and only if
$$\begin{aligned} A=C+D=0,\qquad B=2\varepsilon \alpha , \quad \varepsilon ^2=1. \end{aligned}$$
In order to determine Einstein examples, consider the algebraic system of equations
$$\begin{aligned} \varrho _{ij}=\lambda g_{ij},\quad i,j=1,\dots 4, \end{aligned}$$
which immediately gives the following solutions
$$\begin{aligned} \left\{ \begin{array}{l} A=C+D=0,\quad B=2\varepsilon \alpha \\ (2A-D)=C,\quad C^2=(D^2+8\alpha ^2),\quad B=0. \end{array}\right. \end{aligned}$$
Since the first solution yields to being Ricci flat, so we omit this solution in the Table 1.
Now applying the Eqs. (3.2) and (3.4), the manifold is Ricci parallel if and only if one of the following solutions satisfy
$$\begin{aligned} \left\{ \begin{array}{l} A=C+D=0,\quad B=2\varepsilon \alpha , \\ CA-A^2=A^2-DA=2\alpha ^2,\quad B=0,\\ CA=-DA=\alpha ^2,\quad B=0.\end{array}\right. \end{aligned}$$
Excluding the first and second solutions which are Ricci flat and Einstein respectively, just the third solution is contained in the Table 1 and this finish the proof. \(\square \)

Extra geometric properties which are valuable in the study of geometric properties of different spaces are locally symmetry and conformally flatness. We study these conditions on four dimensional neutral Lie groups in the following theorem.

Theorem 3.2

Let (Gg) be one of the Lie groups listed in the Theorems 2.2 and 2.3, locally symmetric and conformally flat examples are listed in the following Tables 3 and 4.

Proof

Like the proof of Theorem 3.1, we examine the case (a1), and all other examples have been investigated in the similar way. By using Eqs. (3.2) and (3.3) and taking into account \(\alpha \ne 0\), the equation \(\Lambda (R)=0\) gives the following solutions for being locally symmetric, after straight forward calculations.
$$\begin{aligned} \left\{ \begin{array}{l} AC=-AD=\alpha ^2,\quad B=0,\\ AC-A^2=A^2-AD=2\alpha ^2,\quad B=0. \end{array}\right. \end{aligned}$$
Now we set the Eq. (3.1) equal to zero which translates the conformally flat condition. Standard calculations show that for \(\alpha \ne 0\), the manifold is conformally flat if and only if
$$\begin{aligned} AC-A^2=A^2-AD=2\alpha ^2,\quad B=0, \end{aligned}$$
and this matter ends the proof. \(\square \)

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© Sociedade Brasileira de Matemática 2018

Authors and Affiliations

  1. 1.Department of MathematicsPayame Noor UniversityTehranIran
  2. 2.Department of Mathematics, Basic Sciences FacultyUniversity of BonabBonabIran

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