Holonomy groups of ${{G}_{2}^*}$-manifolds

Abstract

Abstract. We classify the holonomy algebras of manifolds admitting an indecomposable torsion free G * 2 -structure, i.e. for which the holonomy representation does not leave invariant any proper non-degenerate subspace. We realize some of these Lie algebras as holonomy algebras of left-invariant metrics on Lie groups.

Local Type I Metrics with Holonomy in G₂^*

“…For Type II we realised n, sl(2, R) 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 G * 2 -structure [7,11]. Furthermore, Willse proved for some of the Berger algebras that they appear as the holonomy of an ambient metric.…”

“…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 m, R · N m and s 1/2 m as holonomy algebras. Furthermore, Willse constructed an ambient metric with holonomy s 1/2 m (personal communication).…”

“…We start by choosing b 5 = dx 5 , b 6 = dx 6 , which implies n ∈ I(dx 5 ). Moreover, we choose b 7 = dx 7 +p(x 5 , x 6 )dx 5 , b 4 = dx 4 , thus v ∈ I(dx 5 , dx 7 ). Furthermore, we may choose b 3 = dx 3 +qdx 6 .…”

“…, x 7 ) = x 2 we get holonomy equal to b 2 m or d m, respectively. Recall that we already gave an example of a metric with holonomy R · N m in [7]. It remains to consider those Berger algebras h = a m for which a is equal to R · S,b 2 , co (2),…”

Holonomy Classification of Lorentz-Kähler Manifolds

“…The case of Lorentzian manifolds took the attention of geometers and theoretical physicists during the last two decades, see the reviews [3,25,26] and the references therein. In the other signatures, only partial results are known [12,13,7,23,24,28,29,35]. The main difference between holonomy groups of Riemannian and proper pseudo-Riemannian manifolds is that for Riemannian manifolds all considerations may be reduced to the case of irreducible holonomy groups, while in the case of proper pseudo-Riemannian manifolds one should consider also holonomy groups preserving degenerate subspaces.…”