Local Type II metrics with holonomy in ${\mathrm {G}}_2^*$

Abstract

A list of possible holonomy groups with indecomposable holonomy representation contained in the exceptional, non-compact Lie group ${\mathrm {G}}_2^{*}$ was provided by Fino and Kath. The classification is due to the corresponding holonomy algebras and divided into Type I, II and III, depending on the dimension of the socle being 1, 2 or 3, respectively. It was also shown by Fino and Kath that all algebras of Type I, and by the author that all of Type III are indeed realizable as holonomy algebras by metrics with signature (4, 3). This article proves that this is also true for all Type II algebras. Thus, there exists a realization by a metric for all indecomposable holonomy groups contained in ${\mathrm {G}}_2^{*}$.

Appendix

Table 2 Local functions used for (a) adapted coframes $(b^i)$. (b) (quasi-)normalforms of the basis $b^i$. Functions not listed are identically zero

The appendix is to read as follows: Table 2 provides how the basis $b^i$ is expressed in terms of local functions $f^j_{i}$ and coordinates $x_i$ on M. Instead of using the general expression

$$\begin{aligned} b^i = {\mathrm {e}}^{f_i^i}{\mathrm {d}}x_i + \sum _{j \ne i, j \ge 4} f^{j}_{i} {\mathrm {d}}x_j \end{aligned}$$

we assign a certain letter to local functions of each $b^i$ as in the example. Thus, we express $b^1,\ldots ,b^7$ as

$$\begin{aligned} b^1&= {\mathrm {d}}x_1+ \sum _{ i \ge 4} r_i \, {\mathrm {d}}x_i\;,\qquad b^2 = {\mathrm {d}}x_2 + \sum _{i \ge 4} s_i \, {\mathrm {d}}x_i\;,\qquad b^3 = {\mathrm {d}}x_3 + \sum _{i \ge 6} t_i \,{\mathrm {d}}x_i\;,\\ b^4&= {\mathrm {d}}x_4 + \sum _{i \ge 6} u_i \, {\mathrm {d}}x_i\;, \qquad b^5 = {\mathrm {d}}x_5 + \sum _{i \ge 6} v_i \, {\mathrm {d}}x_i\;,\qquad b^6 = {\mathrm {e}}^{w_6}\,{\mathrm {d}}x_6 + w_7\, {\mathrm {d}}x_7\;,\\ b^7&= {\mathrm {e}}^{z_7}\,{\mathrm {d}}x_7 {\,}, \end{aligned}$$

where $i \in \{4,\ldots ,7\}$ and $r_i,s_i,t_i,u_i,v_i,w_i,z_i$ are local functions. Note, there are only a few cases where $w_6, z_7 \ne 0$ and we dropped all functions which are not needed further.

In the following subsections local functions which generate the particular holonomy algebra in each case of Theorem 3 are listed. Since dependencies on local coordinates vary from case to case, they are not listed. Instead, they can be extracted in each case from Table 2.

Since the calculations themselves are quite lengthy and somewhat repetitive, they are not given here. Readers interested in details can obtain the Maple files used to compute the holonomy algebras via the author’s website and on request. In case of any questions or problems with these files do not hesitate to contact the author.

Type II 1

Let $\mathfrak {a} \in \{\mathfrak {gl}(2,\mathbb {R}), \mathfrak {sl}(2,\mathbb {R})\}$ and $\mathfrak {h} = \mathfrak {a}\ltimes \mathfrak {n}$.

Type II 2

Let $\mathfrak {a}\in \left\{ \mathfrak {co}(2),\mathbb {R}\cdot C_a \right\}$ and $\mathfrak {h} = \mathfrak {a}\ltimes \mathfrak {n}$ or $\mathfrak {h}=\mathfrak {a}\ltimes Z_0$.

Type II 3

$\mathfrak {a} = \mathfrak {d}$ and $\mathfrak {h} = \mathfrak {a}\ltimes \mathfrak {n_1}$, where $\mathfrak {n}_1 \in \left\{ \mathfrak {n},\mathfrak {n}(1,3),\mathfrak {n}(2,3),\mathfrak {n}(1,2,3),\mathfrak {n}(1,2,4) \right\}$.

Type II 4

Let $\mathfrak {a}=\mathbb {R}\cdot \mathrm {diag}(1,\mu )$.

(a) $\mathfrak {h}=\mathfrak {a}\ltimes \mathfrak {n_1}, \mu \in [-1,1),\gamma =1-\mu$,

where $\mathfrak {n}_1 \in \left\{ \mathfrak {n},\mathfrak {n}(2,3),\mathfrak {n}(1,2,3),\mathfrak {n}(1,2,4),\mathfrak {n}(1,3,4),\mathfrak {n}(2,3,4)\right\}$.

(b) $\mathfrak {h}=h\left( \mathrm {diag}(1,\frac{1}{2}),(1,0,0,0),0\right) \ltimes \mathfrak {n_1}$, where $\mathfrak {n}_1 \in \left\{ \mathfrak {n}(2,3),\mathfrak {n}(2,3,4)\right\}$.

(c) $\mu =0$ and $\mathfrak {h}=\mathfrak {a}\ltimes \mathfrak {n}(2,4)$ or $\mathfrak {h}=\mathbb {R}\cdot h\left( \mathrm {diag}(1,0),(0,1,0,0),0\right) \ltimes \mathfrak {n}_1$, $\mathfrak {n}_1 \in \left\{ \mathfrak {n}(1,4), \mathfrak {n}(3,4),\mathfrak {n}(1,3,4)\right\}$.

Type II 5

(a) Let $\mathfrak {h} = \mathfrak {n}_1$, where $\mathfrak {n}_1 \in \left\{ \mathfrak {n},\mathfrak {n}(1,3),\mathfrak {n}(2,3),\mathfrak {n}(1,2,4),\mathfrak {n}(2,3,4),Z_1,Z_2,Z_3,Z_4,Z_5 \right\}$.

(b) Let $\mathfrak {h} = \mathbb {R}\cdot I \ltimes \mathfrak {n}_1$, where $\mathfrak {n}_1 \in \left\{ \mathfrak {n},\mathfrak {n}(1,3),\mathfrak {n}(2,3),\mathfrak {n}(1,2,4),\mathfrak {n}(2,3,4),Z_1,Z_2,\right. \left. Z_3,Z_4,Z_5 \right\}$.