Local Type I Metrics with Holonomy in ${\rm G}_{2}^*$

TL;DR

This paper demonstrates that all Type I indecomposable holonomy algebras for ${ m G}_{2}^*$-structures can be realized as the holonomy of a local metric, using Cartan's exterior differential systems.

Contribution

It shows that all Type I holonomy algebras from a known list are realizable as local metric holonomies within the maximal parabolic subalgebra.

Findings

All Type I Lie algebras are realizable as local holonomy.

These Lie algebras are contained in the maximal parabolic subalgebra p_1.

The realization is achieved via Cartan's exterior differential systems.

Abstract

By [arXiv:1604.00528], a list of possible holonomy algebras for pseudo-Riemannian manifolds with an indecomposable torsion free ${\rm G}_{2}^*$-structure is known. Here indecomposability means that the standard representation of the algebra on ${\mathbb R}^{4,3}$ does not leave invariant any proper non-degenerate subspace. The dimension of the socle of this representation is called the type of the Lie algebra. It is equal to one, two or three. In the present paper, we use Cartan's theory of exterior differential systems to show that all Lie algebras of Type I from the list in [arXiv:1604.00528] can indeed be realised as the holonomy of a local metric. All these Lie algebras are contained in the maximal parabolic subalgebra $\mathfrak p_1$ that stabilises one isotropic line of ${\mathbb R}^{4,3}$. In particular, we realise $\mathfrak p_1$ by a local metric.

Equations

$$\displaystyle\omega=\sqrt{2}\big{(}e^{167}+e^{235}\big{)}-e^{4}\wedge\big{(}e^{15}-e^{26}-e^{37}\big{)}$$

$$\displaystyle{\langle\cdot\,,\cdot\rangle}=2\big{(}e^{1}\cdot e^{5}+e^{2}\cdot e^{6}+e^{3}\cdot e^{7}\big{)}-\big{(}e^{4}\big{)}^{2}$$

$$\displaystyle{\mathfrak{h}}^{I}:=\big{\{}h(A,v,u,y)\,|\,A\in\mathop{{\mathfrak{g}\mathfrak{l}}}(2,{\mathbb{R}}),\,v\in{\mathbb{R}},\,u,y\in{\mathbb{R}}^{2}\big{\}}\cong{\mathfrak{p}}_{1}$$

$$h(A,v,u,y)=\left(\begin{array}{ccccccc}\mathrm{tr}A& -u_2& u_1& \sqrt{2}v& 0& -y_1& -y_2\\ 0& a_1& a_2& \sqrt{2}{u}_1& y_1& 0& v\\ 0& a_3& a_4& \sqrt{2}{u}_2& y_2& -v& 0\\ 0& 0& 0& 0& \sqrt{2}v& \sqrt{2}{u}_1& \sqrt{2}{u}_2\\ 0& 0& 0& 0& -\mathrm{tr}A& 0& 0\\ 0& 0& 0& 0& u_2& -a_1& -a_3\\ 0& 0& 0& 0& -u_1& -a_2& -a_4\end{array}\right)$$

$$\displaystyle{\mathfrak{m}}:=\big{\{}h(0,v,u,y)\,|\,v\in{\mathbb{R}},\ u,y\in{\mathbb{R}}^{2}\big{\}}\subset{\mathfrak{h}}^{I}$$

$$\displaystyle A\cdot h(0,v,u,y)=h\big{(}0,\operatorname{tr}(A)v,Au,(A+\operatorname{tr}A)y\big{)}.$$

$$\displaystyle[h(0,v,u,y),h(0,\bar{v},\bar{u},\bar{y})]=h\big{(}0,2\theta(u,\bar{u}),0,3(\bar{v}u-v\bar{u})\big{)},$$

$$\displaystyle{\mathfrak{m}}(1,0,0):=\{h(0,v,0,0)\,|\,v\in{\mathbb{R}}\},\qquad{\mathfrak{m}}(0,1,0):=\big{\{}h\big{(}0,0,(u_{1},0)^{\top},0\big{)}\,|\,u_{1}\in{\mathbb{R}}\big{\}},$$

$$\displaystyle{\mathfrak{m}}(0,0,2):=\big{\{}h(0,0,0,y)\,|\,y\in{\mathbb{R}}^{2}\big{\}}.$$

$$\mathfrak{m}(i,j,2)=\mathfrak{m}(i,0,0)\oplus \mathfrak{m}(0,j,0)\oplus \mathfrak{m}(0,0,2)$$

$$C_a:=\left(\begin{array}{cc}a& -1\\ 1& a\end{array}\right),S:=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right),N:=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),$$

$$\displaystyle{\mathfrak{d}}:=\{\operatorname{diag}(a,d)\,|\,a,d\in{\mathbb{R}}\},\qquad\mathop{{\mathfrak{c}\mathfrak{o}}}(2):=\left\{\left(\begin{matrix}a&-b\\ b&a\end{matrix}\right)\Big{|}\,a,b\in{\mathbb{R}}\right\},$$

$${\hat{\mathfrak{b}}}_2:=\mathrm{span}\{I,N\},{\mathfrak{s}}_{\lambda }:=\mathrm{span}\{\mathrm{diag}(\lambda ,\lambda -1),N\},\lambda \in \mathbb{R},$$

$${\mathfrak{b}}_2:=\text{Lie algebra of upper triangular matrices}.$$

$$d\eta =-\theta \wedge \eta$$

$$d\theta =-\theta \wedge \theta +R(\eta \wedge \eta ),$$

$$\displaystyle 0\rightarrow K(\mathfrak{h})\rightarrow S^{2}(\mathfrak{h})\overset{\wedge}{\rightarrow}\Lambda^{4}\big{(}\mathbb{R}^{n}\big{)}.$$

$$\displaystyle\omega=\sqrt{2}\big{(}b^{167}+b^{235}\big{)}-b^{4}\wedge\big{(}b^{15}-b^{26}-b^{37}\big{)}$$

$$\displaystyle g=2\big{(}b^{1}\cdot b^{5}+b^{2}\cdot b^{6}+b^{3}\cdot b^{7}\big{)}-\big{(}b^{4}\big{)}^{2}.$$

$$\left(\begin{array}{c}d{b}^1\\ d{b}^2\\ d{b}^3\\ d{b}^4\\ d{b}^5\\ d{b}^6\\ d{b}^7\end{array}\right)=-\left(\begin{array}{ccccccc}\mathrm{tr}\mathbf{A}& -{\mathbf{u}}_2& {\mathbf{u}}_1& \sqrt{2}\mathbf{v}& 0& -{\mathbf{y}}_1& -{\mathbf{y}}_2\\ 0& {\mathbf{a}}_1& {\mathbf{a}}_2& \sqrt{2}{\mathbf{u}}_1& {\mathbf{y}}_1& 0& \mathbf{v}\\ 0& {\mathbf{a}}_3& {\mathbf{a}}_4& \sqrt{2}{\mathbf{u}}_2& {\mathbf{y}}_2& -\mathbf{v}& 0\\ 0& 0& 0& 0& \sqrt{2}\mathbf{v}& \sqrt{2}{\mathbf{u}}_1& \sqrt{2}{\mathbf{u}}_2\\ 0& 0& 0& 0& -\mathrm{tr}\mathbf{A}& 0& 0\\ 0& 0& 0& 0& {\mathbf{u}}_2& -{\mathbf{a}}_1& -{\mathbf{a}}_3\\ 0& 0& 0& 0& -{\mathbf{u}}_1& -{\mathbf{a}}_2& -{\mathbf{a}}_4\end{array}\right)\wedge \left(\begin{array}{c}{b}^1\\ b^2\\ b^3\\ b^4\\ b^5\\ b^6\\ b^7\end{array}\right)$$

$$d{x}_{(i_1,\dots ,i_k)}:=d{x}_{i_1}\wedge \cdots \wedge d{x}_{i_k}.$$

$$\displaystyle I_{0}:=I(dx_{5},dx_{6},dx_{7}),\qquad I_{1}:=I\big{(}dx_{(5,6)},dx_{(6,7)},dx_{(5,7)}\big{)}.$$

$$b^1=d{x}_1+r_5(x_1,\dots ,x_7)d{x}_5+r_6(x_1,\dots ,x_7)d{x}_6+r_7(x_1,\dots ,x_7)d{x}_7,$$

$$b^2=d{x}_2+q_2(x_2,\dots ,x_7)d{x}_6+s_2(x_2,\dots ,x_7)d{x}_7,$$

$$b^3=d{x}_3+q_3(x_2,\dots ,x_7)d{x}_6+s_3(x_2,\dots ,x_7)d{x}_7,$$

$$b^4=d{x}_4+q_4(x_5,x_6,x_7)d{x}_6,$$

$$b^5=f(x_5,x_6,x_7)d{x}_5,$$

$$b^j=d{x}_j,j=6,7,$$

$$(s_2-q_3{)}_{x_i}=0,i\in \{2,3\},(s_2-q_3{)}_{x_4}=-(q_4{)}_{x_7},$$

$$(r_6{)}_{x_1}=f_{x_6}/f=(q_2{)}_{x_2}+(q_3{)}_{x_3},(r_7{)}_{x_1}=f_{x_7}/f=(q_3{)}_{x_2}+(s_3{)}_{x_3},$$

$$(r_6{)}_{x_2}=-\sqrt{2}(q_3{)}_{x_4},(r_7{)}_{x_2}=-\sqrt{2}(s_3{)}_{x_4},$$

$$(r_6{)}_{x_3}=\sqrt{2}(q_2{)}_{x_4},(r_7{)}_{x_3}=\sqrt{2}((q_3{)}_{x_4}-(q_4{)}_{x_7}),$$

$$(r_6{)}_{x_4}=-2\sqrt{2}\left((s_2{)}_{x_6}-(q_2{)}_{x_7}-\sum _{j=2}^4(s_2{)}_{x_j}{q}_j+\sum _{j=2}^3(q_2{)}_{x_j}{s}_j\right)-(q_4{)}_{x_5}/f,$$

$$(r_7{)}_{x_4}=-2\sqrt{2}\left((s_3{)}_{x_6}-(q_3{)}_{x_7}-\sum _{j=2}^4(s_3{)}_{x_j}{q}_j+\sum _{j=2}^3(q_3{)}_{x_j}{s}_j\right),$$

$$(r_5{)}_{x_1}=-\frac{f}{\sqrt{2}}(q_4{)}_{x_7},$$

$$(r_5{)}_{x_2}=-\frac{f}{2\sqrt{2}}(r_7{)}_{x_4},$$

$$(r_5{)}_{x_3}=-\frac{1}{2\sqrt{2}}(q_4{)}_{x_5}+\frac{f}{2\sqrt{2}}(r_6{)}_{x_4},$$

$$(r_5{)}_{x_4}=\frac{f}{\sqrt{2}}\left((r_6{)}_{x_7}-(r_7{)}_{x_6}+(r_7{)}_{x_1}{r}_6-(r_6{)}_{x_1}{r}_7+\sum _{j=2}^4(r_7{)}_{x_j}{q}_j-\sum _{j=2}^3(r_6{)}_{x_j}{s}_j\right)$$

$$+\frac{1}{\sqrt{2}}((q_3{)}_{x_5}-(s_2{)}_{x_5}).$$

Full text

\FirstPageHeading

\ShortArticleName

Local Type I Metrics with Holonomy in ${\rm G}_{2}^{*}$

\ArticleName

Local Type I Metrics with Holonomy in $\boldsymbol{{\rm G}_{2}^{*}}$

\Author

Anna FINO † and Ines KATH ‡

\AuthorNameForHeading

A. Fino and I. Kath

\Address

† Dipartimento di Matematica G. Peano, Università di Torino,

† Via Carlo Alberto 10, Torino, Italy \EmailDannamaria.fino@unito.it

\Address

‡ Institut für Mathematik und Informatik, Universität Greifswald,

‡ Walther-Rathenau-Str. 47, D-17487 Greifswald, Germany \EmailDines.kath@uni-greifswald.de

\ArticleDates

Received October 24, 2017, in final form July 29, 2018; Published online August 03, 2018

\Abstract

By [arXiv:1604.00528], a list of possible holonomy algebras for pseudo-Riemannian manifolds with an indecomposable torsion free ${\rm G}_{2}^{*}$-structure is known. Here indecomposability means that the standard representation of the algebra on ${\mathbb{R}}^{4,3}$ does not leave invariant any proper non-degenerate subspace. The dimension of the socle of this representation is called the type of the Lie algebra. It is equal to one, two or three. In the present paper, we use Cartan’s theory of exterior differential systems to show that all Lie algebras of Type I from the list in [arXiv:1604.00528] can indeed be realised as the holonomy of a local metric. All these Lie algebras are contained in the maximal parabolic subalgebra $\mathfrak{p}_{1}$ that stabilises one isotropic line of ${\mathbb{R}}^{4,3}$. In particular, we realise $\mathfrak{p}_{1}$ by a local metric.

\Keywords

holonomy; pseudo-Riemannian manifold; exterior differential system; torsion-free G-structures

\Classification

53C29; 53C50; 53C10

1 Introduction

Many problems in geometry can be rephrased as the problem of locally prescribing a given group as holonomy, and this can be reduced to a PDE problem in a number of ways, but most of these lead to PDE that are either degenerate or overdetermined in some way, so the methods of exterior differential systems turn out to be essentially involved. In [4] Bryant showed that the local existence of many non-Riemannian holonomy groups is based on the translation of the structure equations for a given holonomy group into an exterior differential system, and then to use Cartan–Kähler theory [5] to conclude the existence of such metrics. We recall that an exterior differential system is a system of equations on a manifold defined by equating to zero a number of exterior differential forms.

In the present paper, we consider $7$-dimensional pseudo-Riemannian manifolds $(M,g)$ with holonomy $H$ contained in the non-compact subgroup $\mathop{{{\rm G}_{2}^{*}}}\subset\mathop{{\rm SO}}(4,3)$. This is the pseudo-Riemannian analogue of a torsion-free ${\rm G}_{2}$-structure, which is well known from the holonomy theory of Riemannian manifolds since ${\rm G}_{2}$ is one of the groups on Berger’s list [3]. While torsion-free ${\rm G}_{2}$-structures exist on Riemannian 7-manifolds, their pseudo-Riemannian analogues are structures on manifolds of signature $(4,3)$. The Lie group $\mathop{{{\rm G}_{2}^{*}}}$ can be viewed either as the stabiliser of a certain generic 3-form, the stabiliser of a non-isotropic element of the real spinor representation of $\operatorname{Spin}(4,3)$ or the stabiliser of a cross product on ${\mathbb{R}}^{4,3}$. Therefore a torsion-free $\mathop{{{\rm G}_{2}^{*}}}$-structure on a pseudo-Riemannian manifold M of signature $(4,3)$ can be understood as a parallel generic $3$-form, a parallel non-isotropic spinor field or a parallel cross-product $\times$ on M.

Examples of signature (4,3)-metrics with holonomy group equal to $\mathop{{{\rm G}_{2}^{*}}}$ have been constructed, see for instance [2, 6, 8, 9, 12, 14]. In the present paper we are interested in the case where the holonomy is strictly contained in $\mathop{{{\rm G}_{2}^{*}}}$.

In [7] a classification of indecomposable Berger algebras strictly contained in the Lie algebra $\mathfrak{g}_{2}^{*}$ was obtained, where by indecomposable we mean that the holonomy representation does not leave invariant any proper non-degenerate subspace. The indecomposable Berger algebras ${\mathfrak{h}}\subset\mathfrak{g}_{2}^{*}\subset\mathfrak{so}(4,3)$ have been distinguished by the dimension of the socle, i.e., of the maximal semisimple subrepresentation of their natural representation on ${\mathbb{R}}^{4,3}$. As in [7] we will say that ${\mathfrak{h}}$ is of Type I, II or III, if, respectively, the dimension of the socle is one, two or three. The types can be described in terms of the two 9-dimensional maximal parabolic subalgebras $\mathfrak{p}_{1}$, $\mathfrak{p}_{2}$ of $\mathfrak{g}_{2}^{*}$, which can be characterised by the action of $\mathop{{{\rm G}_{2}^{*}}}$ on isotropic subspaces of $\mathbb{R}^{4,3}$. More precisely, $\mathfrak{p}_{1}$ is the Lie algebra of the stabiliser of an isotropic line and $\mathfrak{p}_{2}$ is the stabiliser of an isotropic 2-plane spanned by two vectors $b_{1}$, $b_{2}$ satisfying $b_{1}\times b_{2}=0$. Moreover, $\mathfrak{p}_{1}={\mathfrak{gl}}(2,\mathbb{R})\ltimes\mathfrak{m}$, where $\mathfrak{m}$ is three-step nilpotent, is of Type I and $\mathfrak{p}_{2}={\mathfrak{gl}}(2,\mathbb{R})\ltimes\mathfrak{n}$, where $\mathfrak{n}$ is two-step nilpotent, is of Type II. Both $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$ are indecomposable Berger algebras.

By [7], $\mathfrak{h}$ is contained, up to conjugation, in $\mathfrak{p}_{1}$ if $\mathfrak{h}$ is of Type I or III. If $\mathfrak{h}$ is of Type II, then $\mathfrak{h}\subset\mathfrak{p}_{2}$ up to conjugation. In this way we obtained the list of possible holonomy algebras for pseudo-Riemannian manifolds with a torsion free $\mathop{{{\rm G}_{2}^{*}}}$-structure. To complete the classification of holonomy algebras one must prove that all Berger algebras can be realised as holonomy algebras. In [7], we already did this for some of the Berger algebras, which we realised as holonomy algebras of left-invariant metrics on Lie groups. For Type I, we realised the nilpotent Lie algebra $\mathfrak{m}$ and, furthermore, a 7-dimensional solvable Lie algebra and a 6-dimensional nilpotent Lie algebra. For Type II we realised $\mathfrak{n}$, ${\mathfrak{sl}}(2,\mathbb{R})\ltimes\mathfrak{n}$ and 3-dimensional abelian example, and for Type III a three-dimensional abelian Lie algebra. Moreover, we know which Berger algebras are holonomy algebras of symmetric spaces with an invariant $\mathop{{{\rm G}_{2}^{*}}}$-structure [7, 11]. Furthermore, Willse proved for some of the Berger algebras that they appear as the holonomy of an ambient metric. For instance, ${\mathfrak{n}}$ arises as a holonomy algebra in this way [13].

In the present paper we construct local metrics that have holonomy algebras of Type I. Since the list of Berger algebras of Type I is rather long, we will give all the details only for those Berger algebras that are maximal or minimal in the following sense. We consider the largest Berger algebra of Type I, i.e., the parabolic subalgebra $\mathfrak{p}_{1}$ and we consider all Berger algebras that do not contain all of ${\mathfrak{m}}$. For each Lie algebra ${\mathfrak{h}}$ of this kind we use an adapted coframe to write the structure equations for the $H$-structure and Cartan’s theory of exterior differential systems. This will give a normal form for a metric whose holonomy is contained in the considered Berger algebra ${\mathfrak{h}}$. Then we use this normal form to find a metric whose holonomy is equal to ${\mathfrak{h}}$. For each of the remaining Berger algebras ${\mathfrak{h}}$ from the list we will give only an example of a metric that shows that ${\mathfrak{h}}$ is indeed a holonomy algebra. In this way we prove

Theorem 1.1**.**

Each indecomposable Berger algebra of Type I is the holonomy algebra of a metric of signature $(4,3)$.

Notation. If $b_{1},\dots,b_{n}$ is a basis of a vector space $W$, then we denote by $b^{1},\dots,b^{n}$ its dual basis of $W^{*}$. Furthermore, $b^{i_{1}\dots i_{k}}:=b^{i_{1}}\wedge\dots\wedge b^{i_{k}}\in\bigwedge^{k}W^{*}$.

2 Holonomy representations contained in $\boldsymbol{\mathop{{{\rm G}_{2}^{*}}}}$ and their type

The group $\mathop{{{\rm G}_{2}^{*}}}$ is known to be one of the groups on Berger’s list of the holonomy groups of irreducible pseudo-Riemannian manifolds. In general, the holonomy representation of a pseudo-Riemannian metric is not completely reducible since it can leave invariant isotropic subspaces. Hence one is interested not only in irreducible holonomy representations but also in the much wider class of indecomposable ones, that is, in those whose natural representation on the tangent space does not contain any non-degenerate invariant subspace. Lie algebras of holonomy groups are called holonomy algebras and their natural representation on the tangent space is called holonomy representation. We are interested in this representations rather than in the holonomy algebra as an abstract Lie algebra. Therefore we consider the holonomy algebra of a pseudo-Riemannian manifold of signature $(p,q)$ always as a subalgebra of $\mathop{{\mathfrak{s}\mathfrak{o}}}(p,q)$. Analogously to holonomy groups, it is called indecomposable if it does not leave invariant any non-degenerate subspace of ${\mathbb{R}}^{p,q}$.

This paper is part of a project whose aim is the classification of indecomposable holonomy algebras strictly contained in the Lie algebra ${\mathfrak{g}}^{*}_{2}$ of $\mathop{{{\rm G}_{2}^{*}}}$. Here we want to understand $\mathop{{{\rm G}_{2}^{*}}}$ as the stabiliser of the 3-form

[TABLE]

on ${\mathbb{R}}^{7}$, where $e_{1},\dots,e_{7}$ is a basis of ${\mathbb{R}}^{7}$. This 3-form induces a metric

[TABLE]

of signature $(4,3)$ on ${\mathbb{R}}^{7}$. In particular, we consider $\mathop{{{\rm G}_{2}^{*}}}$ as a subgroup of $\mathop{{\rm SO}}(4,3)$.

In [7], we obtained a classification of indecomposable Berger algebras strictly contained in the Lie algebra ${\mathfrak{g}}^{*}_{2}\subset\mathop{{\mathfrak{s}\mathfrak{o}}}(4,3)$ on ${\mathbb{R}}^{4,3}$. We distinguished three types of such algebras corresponding to the dimension of the maximal semisimple subrepresentation of their natural representation on ${\mathbb{R}}^{4,3}$. This subrepresentation is called socle. If a holonomy algebra ${\mathfrak{h}}$ is indecomposable, then its socle is (totally) isotropic. Hence, for ${\mathfrak{h}}\subset{\mathfrak{g}}_{2}^{*}$ it has dimension one, two or three. Accordingly, we will say that $\mathfrak{h}$ is of Type I, II or III.

Here we want to study the question whether Berger algebras of Type I are indeed holonomy algebras, i.e., whether there are local metrics such that these Berger algebras are the holonomy algebras of these metrics. Let us first recall the classification of indecomposable Berger algebras of Type I.

Let ${\mathfrak{h}}$ be the holonomy algebra of a pseudo-Riemannian manifold $M^{4,3}$ of signature $(4,3)$. Here we always assume that ${\mathfrak{h}}$ is indecomposable. Suppose that the holonomy representation of ${\mathfrak{h}}$ on V:=T_{x_{0}}\big{(}M^{4,3}\big{)} is contained in ${\mathfrak{g}}_{2}^{*}\subset\mathop{{\mathfrak{s}\mathfrak{o}}}(V,{\langle\cdot\,,\cdot\rangle})$, where ${\langle\cdot\,,\cdot\rangle}$ denotes the metric of $M^{4,3}$ at $x_{0}$. If ${\mathfrak{h}}$ is of Type I, then we may choose a basis of $V$ which gives us an identification $V\cong{\mathbb{R}}^{7}$ such that (2.1) and (2.2) hold and, in addition, such that ${\mathfrak{h}}$ is a subalgebra of the maximal parabolic subalgebra

[TABLE]

of ${\mathfrak{g}}_{2}^{*}$, where

[TABLE]

for $A:=\left(\begin{smallmatrix}a_{1}&a_{2}\\ a_{3}&a_{4}\end{smallmatrix}\right)$, $y=(y_{1},y_{2})^{\top}$, $u=(u_{1},u_{2})^{\top}$.

We define

[TABLE]

and identify $\mathop{{\mathfrak{g}\mathfrak{l}}}(2,{\mathbb{R}})$ with $\{h(A,0,0,0)\,|\,A\in\mathop{{\mathfrak{g}\mathfrak{l}}}(2,{\mathbb{R}})\}$. Then ${\mathfrak{h}}^{I}=\mathop{{\mathfrak{g}\mathfrak{l}}}(2,{\mathbb{R}})\ltimes{\mathfrak{m}}$, where $A\in\mathop{{\mathfrak{g}\mathfrak{l}}}(2,{\mathbb{R}})$ acts on ${\mathfrak{m}}$ by

[TABLE]

The Lie bracket on ${\mathfrak{m}}$ is given by

[TABLE]

where $\theta(u,\bar{u}):=u_{1}\bar{u}_{2}-u_{2}\bar{u}_{1}$ for $u,\bar{u}\in{\mathbb{R}}^{2}$. We define subspaces of ${\mathfrak{m}}$ by

[TABLE]

Now we put

[TABLE]

for $i,j\in\{0,1\}$.

Remark 2.1**.**

It is well known that the Lie algebra ${\mathfrak{g}}_{2}^{*}$ has a grading ${\mathfrak{g}}_{2}^{*}=g_{-3}\oplus g_{-2}\oplus g_{-1}\oplus g_{0}\oplus g_{1}\oplus g_{2}\oplus g_{3}$. For details see, e.g., [10, p. 14], where the notation is very similar to ours. The parabolic subalgebra ${\mathfrak{p}}_{1}$ is equal to $g_{0}\oplus g_{1}\oplus g_{2}\oplus g_{3}$ and ${\mathfrak{m}}$ equals $g_{1}\oplus g_{2}\oplus g_{3}$. Moreover, ${\mathfrak{m}}(1,0,0)=g_{2}$, ${\mathfrak{m}}(0,0,2)=g_{3}$ and ${\mathfrak{m}}(0,1,0)$ is a subspace of $g_{1}$.

We define matrices

[TABLE]

and the following Lie subalgebras of $\mathop{{\mathfrak{g}\mathfrak{l}}}(2,{\mathbb{R}})\subset{\mathfrak{h}}^{I}$:

[TABLE]

Let ${\mathfrak{a}}$ be the projection of ${\mathfrak{h}}$ to $\mathop{{\mathfrak{g}\mathfrak{l}}}(2,{\mathbb{R}})\subset{\mathfrak{h}}^{I}$.

Theorem 2.2** ([7]).**

If ${\mathfrak{h}}\subset{\mathfrak{g}}_{2}^{*}$ is an indecomposable Berger algebra of Type I, then there exists a basis of $V$ such that we are in one of the following cases

1. $1)$

{\mathfrak{a}}\in\big{\{}0,\mathop{{\mathfrak{s}\mathfrak{l}}}(2,{\mathbb{R}}),\,\mathop{{\mathfrak{g}\mathfrak{l}}}(2,{\mathbb{R}}),\,\mathop{{\mathfrak{c}\mathfrak{o}}}(2),\,{\mathfrak{b}}_{2},\,\hat{\mathfrak{b}}_{2},\,{\mathfrak{d}},\,{\mathbb{R}}\cdot C_{a},\,{\mathbb{R}}\cdot S\big{\}}* and ${\mathfrak{h}}={\mathfrak{a}}\ltimes{\mathfrak{m}}$,* 2. $2)$

${\mathfrak{a}}={\mathfrak{s}}_{\lambda}=\operatorname{span}\{X:=\operatorname{diag}(\lambda,\lambda-1),\,N\}$* and*

1. $a)$

$\lambda\in{\mathbb{R}}$* and ${\mathfrak{h}}={\mathfrak{a}}\ltimes{\mathfrak{m}}$,* 2. $b)$

$\lambda=1$* and {\mathfrak{h}}={\mathbb{R}}\cdot h\big{(}X,0,(0,1)^{\top},0\big{)}\ltimes({\mathbb{R}}\cdot N\ltimes{\mathfrak{m}}(1,1,2)),* 3. $c)$

$\lambda=2$* and {\mathfrak{h}}=\operatorname{span}\big{\{}X,\,h\big{(}N,0,(0,1)^{\top},0\big{)}\big{\}}\ltimes{\mathfrak{m}}(i,j,2), where $(i,j)\in\{(0,0),(1,0)$, $(1,1)\}$,* 3. $3)$

${\mathfrak{a}}={\mathbb{R}}\cdot\operatorname{diag}(1,\mu)$* and*

1. $a)$

$\mu\in[-1,1]$* and ${\mathfrak{h}}={\mathfrak{a}}\ltimes{\mathfrak{m}}$,* 2. $b)$

$\mu=0$* and {\mathfrak{h}}={\mathbb{R}}\cdot h\big{(}\operatorname{diag}(1,0),0,(0,1)^{\top},0\big{)}\ltimes{\mathfrak{m}}(1,1,2),* 4. $4)$

${\mathfrak{a}}={\mathbb{R}}\cdot N$* and*

1. $a)$

${\mathfrak{h}}={\mathfrak{a}}\ltimes{\mathfrak{m}}$, 2. $b)$

{\mathfrak{h}}={\mathbb{R}}\cdot h\big{(}N,0,(0,1)^{\top},0\big{)}\ltimes{\mathfrak{m}}(1,j,2)* for $j\in\{0,1\}$.*

Our aim is to show that all these Lie algebras are indeed holonomy algebras of a metric of signature $(4,3)$ (in particular, they are all Berger algebras). First we will concentrate on maximal and minimal examples in the following sense. We realise as a holonomy algebra the parabolic subalgebra ${\mathfrak{h}}^{I}\cong\mathfrak{p}_{1}$ and all Berger algebras that do not contain all of ${\mathfrak{m}}$. In all these cases we give a kind of local normal form for a metric whose holonomy is contained in the considered Berger algebra ${\mathfrak{h}}$ and we give an example of a metric with holonomy equal to ${\mathfrak{h}}$. For each of the remaining Berger algebras we will give only an example of a metric that shows that this Lie algebra is a holonomy algebra. In this way we prove Theorem 1.1.

Before we start, we mention that for some of the listed Berger algebras already metrics are known. For instance, in [7], we constructed left-invariant metrics on Lie groups realising ${\mathfrak{m}}$, ${\mathbb{R}}\cdot N\ltimes{\mathfrak{m}}$ and ${\mathfrak{s}}_{1/2}\ltimes{\mathfrak{m}}$ as holonomy algebras. Furthermore, Willse constructed an ambient metric with holonomy ${\mathfrak{s}}_{1/2}\ltimes{\mathfrak{m}}$ (personal communication).

We recall that in general, given an $n$-dimensional pseudo-Riemannian manifold $(M,g)$ with holonomy $H$, the holonomy bundle of the metric $g$ is always a $1$-flat $H$-structure, i.e., there exists a compatible torsion-free connection (see for instance [4]). The structure equations on the $H$-structure $B\rightarrow M$ are given by

[TABLE]

and

[TABLE]

where $\eta\colon TB\rightarrow{\mathbb{R}}^{n}$, $\theta\colon TB\rightarrow\mathfrak{h}$ and $R\colon B\rightarrow K(\mathfrak{h})$ is the curvature function with $\mathfrak{h}$ the Lie algebra of $H$. Here $K(\mathfrak{h})$ denotes the $H$-representation

[TABLE]

Using a local frame $(b_{i})$ trivializing the $1$-flat $H$-structure it is possible to write down locally the structure equations for the 1-flat $H$-structure and reformulate the structure equations as a set of differential equations for local functions on the manifold. Note that in terms of the local frame we have that $\theta$ is given by $\theta^{i}_{j}=b^{i}(\nabla{b_{j}})$, where $\nabla$ is the Levi-Civita connection and the 1-form $b^{i}(\nabla{b_{j}})$ is defined by $b^{i}(\nabla{b_{j}})(X):=b^{i}(\nabla_{X}{b_{j}})$ for all tangent vectors $X$.

In order to show that all the Berger algebras $\mathfrak{h}$ of type I can be realized as holonomy algebras, we first obtain a local normal form for metrics with holonomy algebra contained in the maximal parabolic subalgebra $\mathfrak{p}_{1}$. Let $P_{1}$ be the Lie group with Lie algebra $\mathfrak{p}_{1}$. Since the existence of a 1-flat $P_{1}$-structure is equivalent to the induced pseudo-Riemannian metric having holonomy algebra contained in $\mathfrak{p}_{1}$, we first write down locally the structure equations for a 1-flat $P_{1}$-structure trivialized by an adapted local frame $(b_{i})$ and reformulate these structure equations as a set of differential equations for local functions on the manifold. Then we give a solution to the mentioned differential equations. By using Ambrose–Singer holonomy theorem [1], we compute enough components of the curvature tensor and its covariant derivative to conclude that the holonomy algebra is equal to $\mathfrak{p}_{1}$. We use the same argument for all indecomposable Berger algebras $\mathfrak{h}$ of type I which do not contain all of $\mathfrak{m}$. For the remaining Berger algebras $\mathfrak{h}$ we only give a solution of the above mentioned differential equations which shows that ${\mathfrak{h}}$ is a holonomy algebra.

3 Local metrics with holonomy of Type I

3.1 Adapted coordinates

In this subsection, we introduce a normal form for metrics whose holonomy algebra is contained in ${\mathfrak{h}}^{I}\cong{\mathfrak{p}}_{1}$. We will use this normal form in the next subsection to prove that ${\mathfrak{h}}^{I}$ is indeed a holonomy algebra. Let $b_{1},\dots,b_{7}$ be a local section in the reduction of the frame bundle of $M^{4,3}$ to the holonomy group. With respect to $b_{1},\dots,b_{7}$ the 3-form defining the $\mathop{{{\rm G}_{2}^{*}}}$-structure equals

[TABLE]

and the metric is given by

[TABLE]

Let ${\mathfrak{h}}^{I}$ be defined as in equation (2.3). The dual frame $b^{1},\dots,b^{7}$ satisfies the structure equations

[TABLE]

for ${\mathbf{A}}:=\left(\begin{smallmatrix}{\mathbf{a}}_{1}&{\mathbf{a}}_{2}\\ {\mathbf{a}}_{3}&{\mathbf{a}}_{4}\end{smallmatrix}\right)$, where bold symbols denote 1-forms.

Definition 3.1**.**

Let $x_{1},\dots,x_{7}$ be (local) coordinates on $M^{4,3}$. We introduce the notation

[TABLE]

We denote by $I(\omega^{1},\dots,\omega^{k})$ the algebraic ideal generated by the differential forms $\omega^{1},\dots,\omega^{k}$ and define

[TABLE]

Proposition 3.2**.**

The holonomy of \big{(}M^{4,3},g\big{)} is contained in ${\mathfrak{h}}^{I}$ if and only if there are local coordinates $x_{1},\dots,x_{7}$ such that g=2\big{(}b^{1}\cdot b^{5}+b^{2}\cdot b^{6}+b^{3}\cdot b^{7}\big{)}-\big{(}b^{4}\big{)}^{2} for

[TABLE]

where the functions $r_{5}$, $r_{6}$, $r_{7}$, $q_{2}$, $q_{3}$, $q_{4}$, $s_{2}$, $s_{3}$ and $f$ satisfy the differential equations

[TABLE]

Proof.

First assume that the holonomy is contained in ${\mathfrak{h}}^{I}$. Let $b_{1},\dots,b_{7}$ be a local section in the reduction of the frame bundle to the holonomy group. The structure equations imply that we can introduce coordinates such that $b^{5}=fdx_{5}$ and $b^{6},b^{7}\in I_{0}$. We transform the local frame pointwise (in a smooth way) by a suitable element of

[TABLE]

and, furthermore, by $\exp h(0,0,u,0)$ for a suitable local map $u\colon M^{4,3}\rightarrow{\mathbb{R}}^{2}$ such that

[TABLE]

(possibly changing $f$). Then

[TABLE]

implies ${\mathbf{a}}_{1},\dots,{\mathbf{a}}_{4},{\mathbf{u}}_{1},{\mathbf{u}}_{2}\in I_{0}$. Now

[TABLE]

implies $f=f(x_{5},x_{6},x_{7})$. Furthermore, $db^{4}\in I(dx_{5},dx_{6})$. Hence $b_{4}=dx_{4}+\hat{q}dx_{5}+q_{4}dx_{6}$. Transforming the local frame by $\exp h(0,v,0,0)$ for a suitable local function $v$ we obtain $\hat{q}=0$. Thus

[TABLE]

Since

[TABLE]

we have $q_{4}=q_{4}(x_{5},x_{6},x_{7})$ and ${\mathbf{v}}\in I_{0}$. Now we see from $\operatorname{tr}A,{\mathbf{u}}_{1},{\mathbf{u}}_{2},{\mathbf{v}},b^{6},b^{7}\in I_{0}$ that

[TABLE]

is in $I_{0}$. Thus $b^{1}=dx_{1}+r_{5}dx_{5}+r_{6}dx_{6}+r_{7}dx_{7}$. Similarly, we proceed with $b^{2}$ and $b^{3}$ and obtain

[TABLE]

Transforming the local frame by $\exp h(0,0,0,y)$ for suitable local function $y\colon M^{4,3}\to{\mathbb{R}}^{2}$ we may assume that $\hat{q}_{2}=\hat{q}_{3}=0$. By (3.2), the functions $q_{2}$, $q_{3}$, $s_{2}$, $s_{3}$ do not depend on $x_{1}$. This shows that under the assumption that the holonomy is in ${\mathfrak{h}}^{I}$, we find coordinates $x_{1},\dots,x_{7}$ such that equations (3.1) and (3.3)–(3.8) hold.

Now assume that we have any local metric $g$ given by (3.1) with respect to a local frame $b_{1},\dots,b_{7}$. Denote the holonomy group of $g$ by $H$ and assume that $b_{1},\dots,b_{7}$ is a section in the reduction of the frame bundle to $H$. Assume further that (3.3)–(3.8) hold. Since the connection reduces to the holonomy reduction, the matrix-valued 1-form $\theta=(\theta^{i}_{j})=(b^{i}(\nabla b_{j}))$ takes values in the Lie algebra ${\mathfrak{h}}$ of $H$. Hence ${\mathfrak{h}}$ is contained in ${\mathfrak{h}}^{I}$ if and only if $\theta$ takes values in ${\mathfrak{h}}^{I}$. Thus, by (2.3), the holonomy of $g$ is contained in ${\mathfrak{h}}^{I}$ if and only if

[TABLE]

Note that there are six more equations which have to be satisfied, but these are automatically fulfilled with the ansatz of the adapted coframe before. Using the Koszul formula, we find

[TABLE]

Hence, (3.22) is equivalent to (3.9), (3.13), (3.14) and (3.18). Furthermore, (3.21) is equivalent to (3.16), (3.17) and

[TABLE]

Moreover, (3.20) is equivalent to

[TABLE]

and (3.19) to

[TABLE]

Using equation (3.9), we see that (3.23)–(3.28) is equivalent to (3.10), (3.11), (3.12) and (3.15). ∎

3.2 Realisation of the maximal parabolic subgroup as a holonomy group

It is easy to check that the functions

[TABLE]

satisfy (3.9)–(3.18). Hence the metric defined by (3.1), where $b_{1},\dots,b_{7}$ are given as in Proposition 3.2, has holonomy ${\mathfrak{h}}\subset{\mathfrak{h}}_{I}$.

We want to show that ${\mathfrak{h}}$ is equal to ${\mathfrak{h}}^{I}$. More exactly, we will show that $R_{56}$, $R_{57}$ and $(\nabla_{b_{5}}R)_{56}$ generate ${\mathfrak{h}}^{I}$ as a Lie algebra. Since by the Ambrose–Singer theorem these operators are in ${\mathfrak{h}}\subset{\mathfrak{h}}^{I}$, this will prove the assertion. We will see that it is not necessary to know all components of these operators. For entries that are not needed, we just write a ‘$*$’ (even if these entries are not complicated). We compute

[TABLE]

at $x=0$, where $A_{56}=\left(\begin{smallmatrix}-\frac{1}{\sqrt{2}}&0\\ 0&0\end{smallmatrix}\right)$, $A_{57}=\left(\begin{smallmatrix}0&0\\ -1&0\end{smallmatrix}\right)$, $A^{\prime}_{56}=\left(\begin{smallmatrix}0&-\frac{1}{\sqrt{2}}\\ 1-\frac{1}{2\sqrt{2}}&0\end{smallmatrix}\right)$. In particular, we get ${\mathfrak{a}}=\mathop{{\mathfrak{g}\mathfrak{l}}}(2,{\mathbb{R}})$. Since we do not know apriori that ${\mathfrak{h}}$ is indecomposable, this is not sufficient to prove ${\mathfrak{h}}={\mathfrak{h}}^{I}$. To finish the proof, we will show that

[TABLE]

is equal to ${\mathbb{R}}^{2}$, which implies that ${\mathfrak{h}}$ contains ${\mathfrak{m}}$. Since $R_{56}$ and $R_{57}$ are in ${\mathfrak{h}}$, also

[TABLE]

is in ${\mathfrak{h}}$, thus $(1,0)^{\top}$ is in ${\mathfrak{u}}$. Hence ${\mathfrak{u}}={\mathbb{R}}^{2}$ because of the ${\mathfrak{a}}$-invariance of ${\mathfrak{u}}$.

3.3 Type I 2(b)

We consider the Lie algebra ${\mathfrak{h}}$ spanned by h\big{(}\operatorname{diag}(1,0),0,(0,1)^{\top},0\big{)}, $N$ and ${\mathfrak{m}}(1,1,2)$. The structure equations are now

[TABLE]

Proposition 3.3**.**

The holonomy of \big{(}M^{4,3},g\big{)} is contained in the Lie algebra ${\mathfrak{h}}$ of Type I $2(b)$ if and only if there are local coordinates $x_{1},\dots,x_{7}$ such that g=2\big{(}b^{1}\cdot b^{5}+b^{2}\cdot b^{6}+b^{3}\cdot b^{7}\big{)}-\big{(}b^{4}\big{)}^{2} for

[TABLE]

where the functions $q_{3}$, $r_{5}$, $r_{6}$, $r_{7}$ are of the form

[TABLE]

and $p$, $q_{2}$, $q_{3}$, $\bar{r}_{5}$, $\bar{r}_{6}$, $\bar{r}_{7}$ satisfy the differential equations

[TABLE]

Proof.

The structure equations imply $b^{5}=fdx_{5}$ for some function $f$. Transforming the local frame by \exp h\big{(}\operatorname{diag}(x,0),0,(0,x)^{\top},0\big{)} for a suitable local function $x$ we may assume $b^{5}=dx_{5}$. Hence ${\mathbf{x}}\in I(dx_{5})$, which implies $db^{6}\in I(dx_{5})$. Thus we can introduce $x_{6}$ such that $b^{6}=dx_{6}+pdx_{5}$. Hence $p=p(x_{5},x_{6})$. Furthermore, (3.29) shows $db^{7}\in I(dx_{5},dx_{6})$. Hence we can introduce $x_{7}$ such that $b^{7}=dx_{7}+f_{5}dx_{5}+f_{6}dx_{6}$. Transforming the local frame by \exp h\big{(}n\cdot N,0,(u_{1},0)^{\top},0\big{)} for suitable local functions $u_{1}$, $n$ we may assume $b^{7}=dx_{7}$. By (3.29), we obtain

[TABLE]

which implies ${\mathbf{n}},{\mathbf{u}}_{1}\in I(dx_{5},dx_{6})$. Consequently, $db^{4}\in I(dx_{5})$. Hence we can introduce $x_{4}$ such that $b^{4}=dx_{4}+f_{4}dx_{5}$. Transforming the local frame by $\exp h(0,v,0,0)$ for a suitable local function $v$ we may assume $b^{4}=dx_{4}$. Because of $db^{4}=0$, equation (3.29) shows that ${\mathbf{v}}\in I_{0}$. Again by (3.29), we obtain $db^{3}\in I(dx_{5},dx_{6})$. Transforming by \exp h\big{(}0,0,0,(0,y_{2})^{\top}\big{)} for a suitable local function $y_{2}$ we may assume $b^{3}=dx_{3}+q_{3}dx_{6}$. Then

[TABLE]

thus $q_{3}=q_{3}(x_{5},x_{6},x_{7})$ and ${\mathbf{y}}_{2}\in I(dx_{4})+I_{0}$. Furthermore, (3.29) yields $db^{2}\in I(dx_{5},dx_{6})$. Transforming the local frame by \exp h\big{(}0,0,0,(y_{1},0)^{\top}\big{)} for a suitable function $y_{1}$ we may assume $b^{2}=dx_{2}+q_{2}dx_{6}$, thus

[TABLE]

Hence, we have $q_{2}=q_{2}(x_{3},\dots,x_{7})$ and ${\mathbf{y}}_{1}\in I(dx_{2},\dots,dx_{7})$. Finally, we have $db^{1}\in I_{0}$, thus $b^{1}=dx_{1}+r_{5}dx_{5}+r_{6}dx_{6}+r_{7}dx_{7}$ and (3.29) gives

[TABLE]

Consequently, $r_{5}=r_{5}(x_{1},\dots,x_{7})$, $r_{6}=r_{6}(x_{2},\dots,x_{7})$, $r_{7}=r_{7}(x_{4},\dots,x_{7})$.

Now let the metric $g$ be defined by (3.1) with respect to the local coordinates that we considered above. Then the holonomy of $g$ is contained in ${\mathfrak{h}}$ if and only if $b^{2}(\nabla b_{1})=b^{3}(\nabla b_{1})=b^{3}(\nabla b_{2})=b^{3}(\nabla b_{3})=0$ and

[TABLE]

Note that there are six more equations which have to be satisfied, but these are automatically fulfilled with the ansatz of the adapted coframe before. Using the Koszul formula, we find $b^{2}(\nabla b_{1})=b^{3}(\nabla b_{1})=b^{3}(\nabla b_{2})=b^{3}(\nabla b_{3})=0$ and

[TABLE]

Hence the system (3.35)–(3.37) is equivalent to

[TABLE]

which is equivalent to the claim. ∎

Remark 3.4**.**

The system (3.31)–(3.34) can be easily solved. Obviously, (3.31) and (3.32) imply

[TABLE]

Now choose $p=p(x_{5},x_{6})$, $\bar{r}_{7}=\bar{r}_{7}(x_{5},x_{6},x_{7})$, $\bar{q}_{3}=\bar{q}_{3}(x_{5},x_{6})$ and $q_{2}=q_{2}(x_{3},\dots,x_{7})$ such that $q_{2}$ satisfies (3.38). By integrating (3.31) and (3.32), we determine $\bar{r}_{6}$ up to a function of $x_{5}$, $x_{6}$, $x_{7}$, which can be chosen arbitrarily. In order to show that we can determine $\bar{r}_{5}$ such that (3.33) and (3.34) are satisfied, we have to check that the derivative of the right hand side of (3.33) with respect to $x_{4}$ equals the derivative of the right hand side of (3.34) with respect to $x_{3}$. But this is true since

[TABLE]

and since (3.31) and (3.38) imply

[TABLE]

Hence we can integrate (3.33) and (3.34). This gives $\bar{r}_{5}$ up to a function of $x_{5}$, $x_{6}$, $x_{7}$, which also can be chosen arbitrarily.

Remark 3.5**.**

We will say that a local coframe $b$ on an open subset $U\subset{\mathbb{R}}^{7}$ is ${\mathfrak{h}}$-adapted if $b$ has the form (3.30) and if the metric $g$ given by (3.1) has holonomy contained in ${\mathfrak{h}}$. We have shown that each pseudo-Riemannian manifold of signature (4,3) whose holonomy is contained in ${\mathfrak{h}}$ is locally isometric to some $(U,g)$, where $U$ is an open subset of ${\mathbb{R}}^{7}$ and $g$ is given by (3.1) with respect to an ${\mathfrak{h}}$-adapted coframe. Furthermore, we proved that ${\mathfrak{h}}$-adapted coframes exist and that a general ${\mathfrak{h}}$-adapted coframe depends on 2 functions in 4 variables, 3 functions in 3 variables and 2 functions in 2 variables. The latter statement follows from Remark 3.4. Indeed, in the sense of Cartan–Kähler theory, the contact system associated with the partial differential equation $u_{zz}=u_{xy}$ (for a function $u=u(x,y,z)$) is involutive and its solution depends on 2 functions in 2 variables. Hence the general solution $q=q(x_{3},\dots,x_{7})$ of (3.38) depends on 2 functions in 4 variables. Furthermore, $p$, $\bar{q}_{3}$ and $\bar{r}_{7}$ can be chosen arbitrarily, which gives 2 functions in 2 variables and 1 function in 3 variables. Finally, the determination of $\bar{r}_{6}$ and $\bar{r}_{7}$ as described in Remark 3.4 gives 2 additional arbitrary functions in 3 variables (integration constants).

Example 3.6**.**

Starting with $p(x_{5},x_{6})=x_{6}^{2}$, $q_{2}(x_{3},\dots,x_{7})=x_{3}+x_{4}$, $\bar{q}_{3}=\bar{r}_{7}=0$ we obtain

[TABLE]

Then one computes

[TABLE]

Since these elements generate ${\mathfrak{h}}$ as a Lie algebra, the holonomy algebra coincides with ${\mathfrak{h}}$.

3.4 Type I 2(c), $\boldsymbol{i=j=0}$

We consider

[TABLE]

The structure equations are

[TABLE]

where bold symbols denote 1-forms.

Proposition 3.7**.**

The holonomy of \big{(}M^{4,3},g\big{)} is contained in the Lie algebra ${\mathfrak{h}}$ of Type I $2(c)$ with $i=j=0$, if and only if we can introduce local coordinates $x_{1},\dots,x_{7}$ such that g=2\big{(}b^{1}\cdot b^{5}+b^{2}\cdot b^{6}+b^{3}\cdot b^{7}\big{)}-\big{(}b^{4}\big{)}^{2} for

[TABLE]

where $p$, $q$, $q_{2}$, $q_{3}$, $q_{4}$, $r_{5}$, $r_{6}$, $r_{7}$ are of the form

[TABLE]

for some functions $a=a(x_{5})$, $b=b(x_{5})$, $\bar{p}=\bar{p}(x_{5},x_{6})$, $\bar{q}=\bar{q}(x_{5},x_{6})$, $\bar{q}_{j}=\bar{q}_{j}(x_{5},x_{6})$ and $\bar{r}_{k}=\bar{r}_{k}(x_{5},x_{6},x_{7})$ for $j=2,3,4$, $k=5,6,7$ that satisfy

[TABLE]

Proof.

We can introduce coordinates such that $b^{5}\in\operatorname{span}\{dx_{5}\}$. Changing the local frame $b_{1},\dots,b_{7}$ by $\exp tX$ for a suitable local function $t$ we obtain $b^{5}=dx_{5}$. Then ${\mathbf{x}}\in I(dx_{5})$, hence $db^{6}\in I(dx_{5})$ by (3.39). Consequently, we can introduce $x_{6}$ such that $b^{6}=dx_{6}+t_{1}dx_{5}$ for some function $t_{1}$. Now we change the local frame $b_{1},\dots,b_{7}$ by \exp\big{(}t_{2}\cdot h\big{(}N,0,(0,1)^{\top},0\big{)}\big{)} for a suitable local function $t_{2}$ such that $b^{6}=dx_{6}$. Then ${\mathbf{n}}\in I(dx_{5},dx_{6})$ by (3.39). Thus $db^{7}\in I(dx_{5})$, which gives $b^{7}=dx_{7}+pdx_{5}$ for suitable functions $x_{7}$ and $p$. Then

[TABLE]

implies $p=p(x_{5},x_{6},x_{7})$. By (3.39), we get $db^{4}=-\sqrt{2}{\mathbf{n}}\wedge b^{7}$, thus $db^{4}\in I(dx_{5},dx_{6})$. Hence we can choose $x_{4}$ such that $b^{4}=dx_{4}+qdx_{5}+q_{4}dx_{6}$. Now

[TABLE]

gives $q=q(x_{5},x_{6},x_{7})$ and $q_{4}=q_{4}(x_{5},x_{6},x_{7})$. Again by (3.39), we have $db^{3}\in I(dx_{5},dx_{6})$. Thus we can choose $x_{3}$ such that $b^{3}=dx_{3}+t_{1}dx_{5}+t_{2}dx_{6}$ for some functions $t_{1}$, $t_{2}$. Now we change the local frame $b_{1},\dots,b_{7}$ by $\exp(h(0,0,0,(0,y_{2})^{\top}))$ for a suitable local function $y_{2}$ such that $b^{3}=dx_{3}+q_{3}dx_{6}$. Since

[TABLE]

we have $dq_{3}\in I(dx_{4},dx_{5},dx_{6})$, thus $q_{3}=q_{3}(x_{4},x_{5},x_{6})$. This gives ${\mathbf{y}}_{2}\in I(dx_{3},dx_{4},dx_{5},dx_{6})$. Similarly, $b^{2}=dx_{2}+q_{2}dx_{6}$, where $q_{2}=q_{2}(x_{3},x_{5},x_{6})$. Thus ${\mathbf{y}}_{1}\in I(dx_{2},dx_{3},dx_{5},dx_{6})$. Finally, (3.39) shows that $db^{1}\in I_{0}$. Thus we can choose $x_{1}$ such that $b^{1}=dx_{1}+r_{5}dx_{5}+r_{6}dx_{6}+r_{7}dx_{7}$. Then

[TABLE]

hence $r_{5}$, $r_{6}$, $r_{7}$ are as claimed.

We proceed as in the proof of Proposition 3.2. We compute $b^{2}(\nabla b_{1})=b^{3}(\nabla b_{1})=b^{3}(\nabla b_{2})=0$ and

[TABLE]

The equation ${\mathbf{x}}=\frac{1}{3}b^{1}(\nabla b_{1})=\frac{1}{2}b^{2}(\nabla b_{2})=b^{3}(\nabla b_{3})$ is equivalent to

[TABLE]

Furthermore, ${\mathbf{n}}=-b^{1}(\nabla(b_{2}))=b^{2}(\nabla b_{3})=\frac{1}{\sqrt{2}}b^{3}(\nabla b_{4})$ is equivalent to

[TABLE]

Finally, $b^{1}(\nabla b_{3})=b^{1}(\nabla b_{4})=b^{2}(\nabla b_{4})=b^{2}(\nabla b_{7})=0$ is equivalent to

[TABLE]

Equations (3.40), (3.42) and (3.45) imply $-4p_{x_{7}}=(r_{6})_{x_{2}}=-2(q_{2})_{x_{3}}$. Hence $p_{x_{7}}$ does not depend on $x_{7}$ and $(q_{2})_{x_{3}}$ does not depend on $x_{3}$. Thus

[TABLE]

Now (3.41), (3.44), (3.45) and (3.48) yield

[TABLE]

and (3.40), (3.41), (3.43) and (3.46) give

[TABLE]

Equations (3.49) and (3.42) imply $(q_{4})_{x_{7}}=(q_{3})_{x_{4}}=-\frac{1}{\sqrt{2}}(r_{6})_{x_{2}}=2\sqrt{2}\hat{p}$, thus

[TABLE]

Moreover, (3.48) gives

[TABLE]

Now (3.47) is equivalent to

[TABLE]

and (3.50) is equivalent to

[TABLE]

Putting $\hat{p}(x_{5},x_{6})=a(x_{5})x_{6}+b(x_{5})$, we obtain the assertion. ∎

Remark 3.8**.**

Similarly to Remark 3.5 we can determine the generality of ${\mathfrak{h}}$-adapted coframes. The functions $a$, $b$, $\bar{q}$, $\bar{q}_{2}$, $\bar{q}_{3}$, $\bar{r}_{5}$, $\bar{r}_{6}$ can be chosen arbitrarily. This gives 2 functions in 3 variables, 3 functions in 2 variables and 2 functions in 1 variable. The stepwise determination of $\bar{p}$, $\bar{q}_{4}$ and $\bar{r}_{7}$ gives 1 more function in 2 variables and 2 functions in 1 variable. All in all, a general ${\mathfrak{h}}$-adapted coframe depends on 2 functions in 3 variables, 4 functions in 2 variables and 4 functions in 1 variable.

Example 3.9**.**

For $a(x_{5}):=1$, $\bar{p}(x_{5},x_{6}):=\frac{1}{6}x_{6}^{4}$, $\bar{q}_{4}(x_{5},x_{6}):=\frac{\sqrt{2}}{3}x_{5}x_{6}^{5}$,

[TABLE]

and $b=\bar{q}_{2}=\bar{q}_{3}=\bar{q}=\bar{r}_{5}=\bar{r}_{7}=0$ the holonomy equals ${\mathfrak{h}}$. To show this we determine the operators $R_{56}$, $R_{57}$ and $R_{36}$, which are in ${\mathfrak{h}}$ by Proposition 3.7. As in Section 3.2 we only compute those entries that are needed and write ‘$*$’ for the remaining ones. We obtain

[TABLE]

which generate ${\mathfrak{h}}$ as a Lie algebra. This proves the assertion.

3.5 Type I 2(c), $\boldsymbol{i=1}$, $\boldsymbol{j=0}$

We consider

[TABLE]

The structure equations are

[TABLE]

Proposition 3.10**.**

The holonomy of \big{(}M^{4,3},g\big{)} is contained in the Lie algebra ${\mathfrak{h}}$ of Type I $2(c)$ with $i=1$, $j=0$ if and only if we can introduce local coordinates $x_{1},\dots,x_{7}$ such that g=2\big{(}b^{1}\cdot b^{5}+b^{2}\cdot b^{6}+b^{3}\cdot b^{7}\big{)}-\big{(}b^{4}\big{)}^{2} for

[TABLE]

where $q_{2}$, $q_{3}$, $r_{5}$, $r_{6}$, $r_{7}$ are of the form

[TABLE]

for functions $\bar{q}_{k}=\bar{q}_{k}(x_{5},x_{6},x_{7})$, $k=2,3$, $\bar{r}_{l}=\bar{r}_{l}(x_{5},x_{6},x_{7})$, $l=5,6,7$, satisfying

[TABLE]

Proof.

As for Type I 2(c), $i=j=0$, we can introduce coordinates such that $b^{k}=dx_{k}$, $k=5,6$, $b^{7}=dx_{7}+p(x_{5},x_{6},x_{7})dx_{5}$ and ${\mathbf{x}}\in I(dx_{5})$, ${\mathbf{n}}\in I(dx_{5},dx_{6})$. But now we can assume additionally, that $b^{4}=dx_{4}+q_{4}dx_{6}$. Then

[TABLE]

gives $q_{4}=q_{4}(x_{5},x_{6},x_{7})$ and ${\mathbf{v}}\in I_{0}$. Again by (3.51), we have $db^{3}\in I(dx_{5},dx_{6})$. As for Type I 2(c), $i=j=0$ we can choose $x_{3}$ such that $b^{3}=dx_{3}+q_{3}dx_{6}$. Since

[TABLE]

we have $q_{3}=q_{3}(x_{4},\dots,x_{7})$ and ${\mathbf{y}}_{2}\in I(dx_{3},\dots,dx_{7})$. Similarly, $b^{2}=dx_{2}+q_{2}dx_{6}$, where $q_{2}=q_{2}(x_{3},x_{5},x_{6},x_{7})$. Thus ${\mathbf{y}}_{1}\in I(dx_{2},dx_{3})+I_{0}$. Finally, (3.51) shows that $db^{1}\in I_{0}$. Thus we can choose $b^{1}=dx_{1}+r_{5}dx_{5}+r_{6}dx_{6}+r_{7}dx_{7}$. Then

[TABLE]

hence $r_{5}$, $r_{6}$, $r_{7}$ are as claimed. We proceed as in the proof of Proposition 3.2. For $\nabla b_{1}=\nabla b_{2}=\nabla b_{3}=0$, we get the same formulas as in the case of Type I 2(c), $i=j=0$. Furthermore,

[TABLE]

The equation ${\mathbf{x}}=\frac{1}{3}b^{1}(\nabla b_{1})=\frac{1}{2}b^{2}(\nabla b_{2})=b^{3}(\nabla b_{3})$ is equivalent to (3.40). Furthermore, ${\mathbf{n}}=-b^{1}(\nabla(b_{2}))=b^{2}(\nabla b_{3})=\frac{1}{\sqrt{2}}b^{3}(\nabla b_{4})$ is equivalent to

[TABLE]

and ${\mathbf{v}}=\frac{1}{\sqrt{2}}b^{1}(\nabla b_{4})=b^{2}(\nabla b_{7})$ is equivalent to

[TABLE]

Finally, $b^{1}(\nabla b_{3})=b^{2}(\nabla b_{4})=0$ is equivalent to

[TABLE]

Equations (3.40) and (3.52)–(3.59) simplify to

[TABLE]

These equations imply, in particular, $2p_{x_{7}x_{7}}=(q_{2})_{x_{3}x_{7}}=(q_{2})_{x_{7}x_{3}}=((q_{4})_{x_{5}}-2\sqrt{2}p_{x_{7}}p)_{x_{3}}=0$. Now the second part of the proposition follows easily. ∎

Remark 3.11**.**

In the same way as in Remarks 3.5 and 3.8 we determine the generality of ${\mathfrak{h}}$-adapted coframes. The functions $\bar{r}_{5}$, $\bar{r}_{6}$, $\bar{r}_{7}$ can be chosen arbitrarily. This gives 3 functions in 3 variables. Integration of $p_{x_{7}x_{7}}=0$ shows that the choice of $p$ depends on 2 functions in 2 variables. Then $\bar{q}_{3}$, $q_{4}$ and $\bar{q}_{2}$ depend on 3 arbitrary functions in 2 variables. Thus a general ${\mathfrak{h}}$-adapted coframe depends on 3 functions in 3 variables and 5 functions in 2 variables.

Example 3.12**.**

Starting with $p(x_{5},x_{6},x_{7})=x_{6}x_{7}$, $\bar{q}_{2}(x_{5},x_{6},x_{7})=-x_{6}^{2}x_{7}^{2}$, $\bar{q}_{3}(x_{5},x_{6},x_{7})=x_{5}-\frac{1}{2}x_{7}^{2}$, $q_{4}(x_{5},x_{6},x_{7})=2\sqrt{2}x_{6}x_{7}$ and $\bar{r}_{5}=\bar{r}_{6}=\bar{r}_{7}=0$ we get

[TABLE]

We want to show that the holonomy equals ${\mathfrak{h}}$. We proceed as in Example 3.9 and compute the following parts of the curvature tensor and its covariant derivative:

[TABLE]

which generate ${\mathfrak{h}}$ as a Lie algebra.

3.6 Type I 2(c), $\boldsymbol{i=j=1}$

We consider

[TABLE]

The structure equations are

[TABLE]

Proposition 3.13**.**

The holonomy of \big{(}M^{4,3},g\big{)} is contained in the Lie algebra ${\mathfrak{h}}$ of Type I $2(c)$, $i=j=1$ if and only if we can introduce coordinates $(x_{1},\dots,x_{7})$ such that g=2\big{(}b^{1}\cdot b^{5}+b^{2}\cdot b^{6}+b^{3}\cdot b^{7}\big{)}-\big{(}b^{4}\big{)}^{2} for

[TABLE]

where $q_{2}$, $q_{3}$, $s$, $r_{5}$, $r_{6}$, $r_{7}$ are of the form

[TABLE]

for arbitrary functions $\bar{s}=\bar{s}(x_{5},x_{6},x_{7})$, $\bar{q}_{k}=\bar{q}_{k}(x_{5},x_{6},x_{7})$, $k=2,3$, $q_{4}=q_{4}(x_{5},x_{6},x_{7})$, $\bar{r}_{l}=\bar{r}_{l}(x_{5},x_{6},x_{7})$, $l=5,6,7$ and $a:=(q_{4})_{x_{7}}$, $b:=(\bar{q}_{3})_{x_{7}}$.

Proof.

We proceed as in the proof of Proposition 3.10 and obtain now $b^{k}=dx_{k}$, $k=5,6,7$, and ${\mathbf{x}}\in I(dx_{5})$, ${\mathbf{n}}\in I(dx_{5},dx_{6})$, ${\mathbf{u}}_{1}\in I_{0}$. Moreover, as above, $b^{4}=dx_{4}+q_{4}dx_{6}$, which implies

[TABLE]

Hence $q_{4}=q_{4}(x_{5},x_{6},x_{7})$ and ${\mathbf{v}}\in I_{0}$. Furthermore, we have $db^{3}\in I(dx_{5},dx_{6})$ and can choose $x_{3}$ such that $b^{3}=dx_{3}+q_{3}dx_{6}$. As in the proof of Proposition 3.10, we see that $q_{3}=q_{3}(x_{4},\dots,x_{7})$ and ${\mathbf{y}}_{2}\in I(dx_{3},\dots,dx_{7})$. Now (3.60) gives $db^{2}\in I_{0}$. Thus $b^{2}=dx_{2}+q_{2}dx_{6}+sdx_{7}$. Since

[TABLE]

we obtain $q_{2}=q_{2}(x_{3},\dots,x_{7})$, $s=s(x_{4},\dots,x_{7})$ and ${\mathbf{y}}_{1}\in I(dx_{2},\dots,dx_{7})$. Finally, (3.60) shows that $db^{1}\in I_{0}$. Thus we can choose $b^{1}=dx_{1}+r_{5}dx_{5}+r_{6}dx_{6}+r_{7}dx_{7}$. Then $db^{1}$ equals

[TABLE]

hence $r_{5}$, $r_{6}$, $r_{7}$ are as claimed.

We proceed as in the proof of Proposition 3.2. The equation ${\mathbf{x}}=\frac{1}{3}b^{1}(\nabla b_{1})=\frac{1}{2}b^{2}(\nabla b_{2})=b^{3}(\nabla b_{3})$ is equivalent to

[TABLE]

Furthermore, ${\mathbf{n}}=-b^{1}(\nabla(b_{2}))=b^{2}(\nabla b_{3})=\frac{1}{\sqrt{2}}b^{3}(\nabla b_{4})$ is equivalent to

[TABLE]

and ${\mathbf{u}}_{1}=b^{1}(\nabla b_{3})=\frac{1}{\sqrt{2}}b^{2}(\nabla b_{4})$ is equivalent to

[TABLE]

Finally, ${\mathbf{v}}=\frac{1}{\sqrt{2}}b^{1}(\nabla b_{4})=b^{2}(\nabla b_{7})$ is equivalent to

[TABLE]

Equations (3.61), (3.63), (3.66) and (3.67) are equivalent to

[TABLE]

Equations (3.62), (3.65) and (3.70) are equivalent to

[TABLE]

Hence we have to solve the system consisting of equations (3.64), (3.68), (3.69), (3.71) and (3.72). This is done in a straightforward way starting with (3.71) and (3.72), continuing with (3.69) and finishing with (3.64) and (3.68). The result shows that $q_{k}$, $k=2,3,4$, $s$ and $r_{l}$, $l=5,6,7$ have the claimed form. ∎

Remark 3.14**.**

Proceeding similarly as in the previous cases we see that a general ${\mathfrak{h}}$-adapted coframe depends on 7 functions in 3 variables.

Example 3.15**.**

Choosing $q_{4}(x_{5},x_{6},x_{7})=x_{6}x_{7}$, $\bar{q}_{3}(x_{5},x_{6},x_{7})=x_{6}x_{7}+\frac{\sqrt{2}}{3}x_{7}^{2}$ and $\bar{s}=\bar{q}_{2}=\bar{r}_{5}=\bar{r}_{6}=\bar{r}_{7}=0$ we obtain

[TABLE]

One computes

[TABLE]

These operators generate ${\mathfrak{h}}$ as a Lie algebra. Hence the holonomy is equal to ${\mathfrak{h}}$.

3.7 Type I 3(b)

We consider

[TABLE]

The structure equations are

[TABLE]

where bold symbols denote 1-forms.

Proposition 3.16**.**

The holonomy of \big{(}M^{4,3},g\big{)} is contained in the Lie algebra ${\mathfrak{h}}$ of Type I $3(b)$ if and only if there are local coordinates $x_{1},\dots,x_{7}$ such that g=2\big{(}b^{1}\cdot b^{5}+b^{2}\cdot b^{6}+b^{3}\cdot b^{7}\big{)}-\big{(}b^{4}\big{)}^{2} for

[TABLE]

where $p=p(x_{5},x_{6})$ and $q_{2}=q_{2}(x_{5},x_{6},x_{7})$ are arbitrary and the functions $q_{3}$, $r_{5}$, $r_{6}$, $r_{7}$ are of the form

[TABLE]

for arbitrary functions $\bar{q}_{3}=\bar{q}_{3}(x_{5},x_{6})$ and $\hat{r}_{j}=\hat{r}_{j}(x_{5},x_{6},x_{7})$, $j=5,6,7$.

Proof.

We can introduce coordinates such that $b^{5}\in\operatorname{span}\{dx_{5}\}$. Transforming the local frame by \exp h\big{(}\operatorname{diag}(x,0),0,(0,x)^{\top},0\big{)} for a suitable local function $x$ we may assume $b^{5}=dx_{5}$. Then ${\bf a}_{1}\in I(dx_{5})$, hence $db^{6}\in I(dx_{5})$ by (3.73). Thus we can introduce $x_{6}$ such that $b^{6}=dx_{6}+pdx_{5}$. Hence $p=p(x_{5},x_{6})$. Furthermore, (3.73) shows $db^{7}\in I(dx_{5})$. Hence we can introduce $x_{7}$ such that $b^{7}=dx_{7}+f_{5}dx_{5}$. Transforming the local frame by $\exp h(0,(u_{1},0)^{\top},0,0)$ for a suitable local function $u_{1}$ we may assume $b^{7}=dx_{7}$. By

[TABLE]

we obtain that ${\bf u}_{1}\in I(dx_{5})$. Consequently, $db^{4}\in I(dx_{5})$. Hence we can introduce $x_{4}$ such that $b^{4}=dx_{4}+f_{4}dx_{5}$. Transforming the local frame by $\exp h(0,v,0,0)$ for a suitable local function $v$ we may assume $b^{4}=dx_{4}$. By

[TABLE]

we obtain that ${\bf v}\in I_{0}$. Again by (3.73), we obtain $db^{3}\in I(dx_{5},dx_{6})$. Transforming the local frame by \exp h\big{(}0,0,0,(0,y_{2})^{\top}\big{)} for a suitable function $y_{2}$ we may assume $b^{3}=dx_{3}+q_{3}dx_{6}$. Then

[TABLE]

thus $q_{3}=q_{3}(x_{5},x_{6},x_{7})$ and ${\mathbf{y}}_{2}\in I(dx_{4})+I_{0}$. Furthermore, (3.73) yields $db^{2}\in I(dx_{5},dx_{6})$. Transforming the local frame by \exp h\big{(}0,0,0,(y_{1},0)^{\top}\big{)} for a suitable function $y_{1}$ we may assume $b^{2}=dx_{2}+q_{2}dx_{6}$, thus

[TABLE]

Hence, we get $q_{2}=q_{2}(x_{5},x_{6},x_{7})$ and ${\mathbf{y}}_{1}\in I(dx_{2},dx_{4})+I_{0}$. Finally, we have $db^{1}\in I_{0}$, thus $b^{1}=dx_{1}+r_{5}dx_{5}+r_{6}dx_{6}+r_{7}dx_{7}$ and (3.73) gives

[TABLE]

Consequently, $r_{5}=r_{5}(x_{1},\dots,x_{7})$, $r_{6}=r_{6}(x_{2},x_{4},x_{5},x_{6},x_{7})$, $r_{7}=r_{7}(x_{4},\dots,x_{7})$.

Now let the metric $g$ be defined by (3.1) with respect to the local coordinates that we considered above. Since the expression for $b^{i}$ in the local coordinates $(x_{1},\ldots,x_{7})$ is the same as for the case Type I 2(b) with $r_{6}=r_{6}(x_{2},x_{4},\ldots,x_{7})$ and $q_{2}=q_{2}(x_{5},x_{6},x_{7})$, we can proceed as in the proof of Proposition 3.3 by imposing the extra conditions coming from

[TABLE]

and $(r_{6})_{x_{3}}=(q_{2})_{x_{3}}=(q_{2})_{x_{4}}=0$. From equations (3.31)–(3.34) we now obtain the system

[TABLE]

By integrating the first equation we determine $\bar{r}_{6}$ up to an arbitrary function $\hat{r}_{6}(x_{5},x_{6},x_{7})$ and, by integrating the other two equations, we get the function $\bar{r}_{5}$, up to an arbitrary function $\hat{r}_{5}(x_{5},x_{6},x_{7})$. ∎

Remark 3.17**.**

Proceeding similarly as in the previous cases we see that in this case a general ${\mathfrak{h}}$-adapted coframe depends on 4 functions in 3 variables and 2 functions in 2 variables.

Example 3.18**.**

For

[TABLE]

the holonomy equals $\mathfrak{h}$. Indeed, the elements

[TABLE]

generate ${\mathfrak{h}}$ as a Lie algebra, the holonomy algebra coincides with ${\mathfrak{h}}$.

3.8 Type I 4(b), $\boldsymbol{j=0}$

We consider

[TABLE]

The structure equations are

[TABLE]

Proposition 3.19**.**

The holonomy of \big{(}M^{4,3},g\big{)} is contained in the Lie algebra ${\mathfrak{h}}$ of Type I $4(b)$ with $j=0$ if and only if there are local coordinates $x_{1},\dots,x_{7}$ such that g=2\big{(}b^{1}\cdot b^{5}+b^{2}\cdot b^{6}+b^{3}\cdot b^{7}\big{)}-\big{(}b^{4}\big{)}^{2} for

[TABLE]

where $p=p(x_{5},x_{6})$ is arbitrary and the functions $q$, $r_{5}$, $r_{6}$, $r_{7}$ are of the form

[TABLE]

where $\bar{q}=\bar{q}(x_{5},x_{6})$, $\bar{r}_{j}=\bar{r}_{j}(x_{5},x_{6},x_{7})$, $j=5,6,7$, are arbitrary.

Proof.

We proceed as in the previous cases integrating the structure equations (3.74). We start by choosing $b^{5}=dx_{5}$, $b^{6}=dx_{6}$, which implies ${\mathbf{n}}\in I(dx_{5})$. Moreover, we choose $b^{7}=dx_{7}+p(x_{5},x_{6})dx_{5}$, $b^{4}=dx_{4}$, thus ${\mathbf{v}}\in I(dx_{5},dx_{7})$. Furthermore, we may choose $b^{3}=dx_{3}+qdx_{6}$. Then

[TABLE]

which yields $q=q(x_{5},x_{6},x_{7})$ and ${\mathbf{y}}_{2}\in I(dx_{4})+I_{0}$. Next we choose $b^{2}=dx_{2}$, which gives ${\mathbf{y}}_{1}\in I(dx_{3})+I_{0}$. Finally, $b^{1}=dx_{1}+r_{5}dx_{5}+r_{6}dx_{6}+r_{7}dx_{7}$. Then

[TABLE]

hence $r_{5}=r_{5}(x_{2},x_{4},x_{5},x_{6},x_{7})$, $r_{6}=r_{6}(x_{3},x_{5},x_{6},x_{7})$, $r_{7}=r_{7}(x_{4},x_{5},x_{6},x_{7})$.

Now let the metric $g$ be defined by (3.1) with respect to the local coordinates that we considered above. Then $\nabla b_{1}=0$ and

[TABLE]

Hence the holonomy of $g$ is contained in ${\mathfrak{h}}$ if and only if

[TABLE]

which is equivalent to the assertion. ∎

Remark 3.20**.**

Proceeding similarly as in the previous cases we see that a general ${\mathfrak{h}}$-adapted coframe depends on 2 functions in 2 variables and 3 functions in 3 variables.

Example 3.21**.**

Starting with $p(x_{5},x_{6})=x_{5}x_{6}+x_{6}^{2}$, $\bar{q}=\bar{r}_{5}=0$, $\bar{r}_{6}(x_{5},x_{6},x_{7})=x_{5}^{2}$ and $\bar{r}_{7}(x_{5},x_{6},x_{7})=x_{6}^{2}$ we obtain

[TABLE]

Then one computes

[TABLE]

Since these elements generate ${\mathfrak{h}}$ as a Lie algebra, the holonomy algebra coincides with ${\mathfrak{h}}$.

3.9 Type I 4(b), $\boldsymbol{j=1}$

We consider

[TABLE]

The structure equations are

[TABLE]

Proposition 3.22**.**

The holonomy of \big{(}M^{4,3},g\big{)} is contained in the Lie algebra ${\mathfrak{h}}$ of Type I $4(b)$ with $j=1$ if and only if there are local coordinates $x_{1},\dots,x_{7}$ such that g=2\big{(}b^{1}\cdot b^{5}+b^{2}\cdot b^{6}+b^{3}\cdot b^{7}\big{)}-\big{(}b^{4}\big{)}^{2} for

[TABLE]

where $q_{3}$ is arbitrary and the functions $q_{2}$ and $r_{5}$, $r_{6}$, $r_{7}$ are of the form

[TABLE]

where $\bar{q}_{2}=\bar{q}_{2}(x_{5},x_{6},x_{7})$, $\bar{r}_{j}=\bar{r}_{j}(x_{5},x_{6},x_{7})$, $j=5,6,7$, are arbitrary.

Proof.

We integrate (3.75). We may choose $b^{j}=dx_{j}$, $j=5,6,7$, which implies ${\mathbf{n}}\in I(dx_{5})$ and ${\mathbf{u}}_{1}\in I(dx_{5},dx_{6})$. Now (3.75) gives $db^{4}\in I(dx_{5})$. Hence we may choose $b^{4}=dx_{4}$. Then ${\mathbf{v}}\in I_{0}$. Again using (3.75) we see that we may choose $b^{k}=dx_{k}+q_{k}dx_{6}$, $k=2,3$. Because of

[TABLE]

we obtain $q_{3}=q_{3}(x_{5},x_{6},x_{7})$ and ${\mathbf{y}}_{2}\in I(dx_{4})+I_{0}$. Furthermore, we have

[TABLE]

This implies $q_{2}=q_{2}(x_{4},x_{5},x_{6},x_{7})$ and ${\mathbf{y}}_{1}\in I(dx_{3},dx_{4})+I_{0}$. Finally, (3.75) shows $db^{1}\in I_{0}$, thus $b^{1}=dx_{1}+r_{5}dx_{5}+r_{6}dx_{6}+r_{7}dx_{7}$. Then

[TABLE]

hence $r_{5}=r_{5}(x_{2},\dots,x_{7})$, $r_{6}=r_{6}(x_{3},\dots,x_{7})$, $r_{7}=r_{7}(x_{4},\dots,x_{7})$.

Let the metric $g$ be defined by (3.1) with respect to the local coordinates that we considered above. Then $\nabla b_{1}=0$ and

[TABLE]

Hence the holonomy of $g$ is contained in ${\mathfrak{h}}$ if and only if

[TABLE]

which is equivalent to the assertion. ∎

Remark 3.23**.**

Proceeding similarly as in the previous cases we see that in this case a general ${\mathfrak{h}}$-adapted coframe depends on 5 functions in 3 variables.

Example 3.24**.**

Starting with $q_{3}(x_{5},x_{6},x_{7})=(x_{5}+x_{6}+x_{7})x_{7}$, $\bar{q}_{2}(x_{5},x_{6},x_{7})=2x_{6}x_{7}$, $\bar{r}_{5}=\bar{r}_{6}=\bar{r}_{7}=0$, we obtain

[TABLE]

To show that the holonomy is equal to ${\mathfrak{h}}$ we again compute parts of the curvature tensor:

[TABLE]

which generate ${\mathfrak{h}}$ as a Lie algebra.

3.10 Holonomy algebras containing $\boldsymbol{{\mathfrak{m}}}$

In the previous sections we concentrated on holonomy groups that are either maximal, i.e., isomorphic to $\mathop{{\mathfrak{g}\mathfrak{l}}}(2,{\mathbb{R}})\ltimes{\mathfrak{m}}$ or that are small in the sense that they do not contain the whole Lie algebra ${\mathfrak{m}}$. For the sake of completeness we consider now also the remaining Berger algebras ${\mathfrak{h}}$ of Type I satisfying ${\mathfrak{m}}\subset{\mathfrak{h}}$. We will not give normal forms for these metrics here, but we will give an example of a metric for each of these Lie algebras. This will complete the proof of Theorem 1.1.

In all cases we will proceed as follows. Given a Berger algebra ${\mathfrak{h}}={\mathfrak{a}}\ltimes{\mathfrak{m}}$, we choose functions $r_{5}$, $r_{6}$, $r_{7}$, $q_{2}$, $q_{3}$, $q_{4}$, $s_{2}$, $s_{3}$ and $f$ that satisfy the differential equations (3.9)–(3.18) and in addition the differential equations for $b^{i}(\nabla b_{j})$ that are equivalent to the condition that the projection of the connection form to $\mathop{{\mathfrak{g}\mathfrak{l}}}(2,{\mathbb{R}})$ is contained in ${\mathfrak{a}}$. This will ensure that the holonomy of the corresponding metric is contained in ${\mathfrak{h}}$. Finally, in each case, one has to check that the holonomy is equal to ${\mathfrak{h}}$. This is done in the same way as for the various examples in Sections 3.2–3.9. If ${\mathfrak{a}}$ is equal to $\mathop{{\mathfrak{c}\mathfrak{o}}}(2)$, ${\mathfrak{b}}_{2}$, ${\mathfrak{d}}$, ${\mathbb{R}}\cdot C_{a}$, ${\mathbb{R}}\cdot S$ or ${\mathbb{R}}\cdot\operatorname{diag}(1,\mu)$ for $\mu\not=0$, the calculations are easy since ${\mathfrak{h}}$ is already generated (as a Lie algebra) by $\{R_{ij}\,|\,i<j\}$. If ${\mathfrak{a}}$ equals $\mathop{{\mathfrak{s}\mathfrak{l}}}(2,{\mathbb{R}})$, $\hat{b}_{2}$, ${\mathbb{R}}\cdot\operatorname{diag}(1,0)$ or ${\mathfrak{s}}_{\lambda}$, then we need also $\nabla R$ to generate ${\mathfrak{h}}$.

According to the described approach we now give functions $f$, $s_{2}$, $s_{3}$, $q_{2}$, $q_{3}$, $q_{4}$, $r_{5}$, $r_{6}$, $r_{7}$ for every ${\mathfrak{a}}\subset\mathop{{\mathfrak{g}\mathfrak{l}}}(2,{\mathbb{R}})$ for which ${\mathfrak{a}}\ltimes{\mathfrak{m}}$ is on the list in Theorem 2.2. We start with ${\mathfrak{a}}=\mathop{{\mathfrak{s}\mathfrak{l}}}(2,{\mathbb{R}})$:

[TABLE]

We continue with ${\mathfrak{a}}\in\{{\mathfrak{b}}_{2},{\mathfrak{d}}\}$. Starting with the functions

[TABLE]

and choosing respectively $q_{2}(x_{2},\dots,x_{7})=x_{2}+e^{x_{3}}$ or $q_{2}(x_{2},\dots,x_{7})=x_{2}$ we get holonomy equal to ${\mathfrak{b}}_{2}\ltimes{\mathfrak{m}}$ or ${\mathfrak{d}}\ltimes{\mathfrak{m}}$, respectively.

Recall that we already gave an example of a metric with holonomy ${\mathbb{R}}\cdot N\ltimes{\mathfrak{m}}$ in [7]. It remains to consider those Berger algebras ${\mathfrak{h}}={\mathfrak{a}}\ltimes{\mathfrak{m}}$ for which ${\mathfrak{a}}$ is equal to ${\mathbb{R}}\cdot S$, $\hat{\mathfrak{b}}_{2}$, $\mathop{{\mathfrak{c}\mathfrak{o}}}(2)$, ${\mathbb{R}}\cdot C_{a}$, ${\mathbb{R}}\cdot\operatorname{diag}(1,\mu)$, $\mu\in[-1,1]$ or to ${\mathfrak{s}}_{\lambda}$, $\lambda\in{\mathbb{R}}$. We may assume $\lambda\not=1/2$ since we considered the case $\lambda=1/2$ already in [7].

For all these ${\mathfrak{a}}$ we choose $f(x_{5},x_{6},x_{7})=e^{x_{5}}$. Furthermore,

[TABLE]

where

[TABLE]

Moreover,

[TABLE]

In order to define $r_{5}$, $r_{6}$, $r_{7}$, we put

[TABLE]

Then, for ${\mathfrak{a}}={\mathbb{R}}\cdot S$,

[TABLE]

For ${\mathfrak{a}}=\hat{\mathfrak{b}}_{2}$,

[TABLE]

If ${\mathfrak{a}}={\mathbb{R}}\cdot\operatorname{diag}(1,\mu)$, $\mu\not=-1$, then

[TABLE]

For ${\mathfrak{a}}={\mathbb{R}}\cdot\operatorname{diag}(1,-1)$,

[TABLE]

If ${\mathfrak{a}}={\mathfrak{s}}_{\lambda}$, $\lambda\not=1/2$, then

[TABLE]

For ${\mathfrak{a}}=\mathop{{\mathfrak{c}\mathfrak{o}}}(2)$,

[TABLE]

and for ${\mathfrak{a}}={\mathbb{R}}\cdot C_{a}$, we put

[TABLE]

Acknowledgements

We thank the referees for their valuable and constructive comments.