Ricci-flat holonomy: A classification

Abstract

The reductive holonomy algebras for a torsion-free affine connection are analysed, with the goal of establishing which ones can correspond to a Ricci-flat connection with the same properties. Various families of holonomies are eliminated through different algebraic means, and examples are constructed (in this paper and in 'Projective Geometry II: Holonomy Classification', by the same author) in the remaining cases, thus solving this problem completely, for reductive holonomy.

Projective holonomy II: cones and complete classifications

“…This connection is Ricci-flat and torsion-free; thus we may appeal to paper [Arm3] which, building on [MeSc1], gives all possible reductive holonomies for Ricci-flat torsion-free affine connections, and use various tricks and theorems to construct either Ricci-flat cones with the required holonomies, or projective manifolds with the required properties.…”

Holonomy algebras of Einstein pseudo-Riemannian manifolds

“…If a 2 + b 2 = 0, then such a form is non-degenerate and it defines the same Levi-Civita connection as g. Thus the metric on M may be chosen to be Einstein and not Ricci-flat. 4. The case of g ⊂ so(n, n) preserving two complementary isotropic subspaces Let g ⊂ so(n, n) and suppose that g preserves two complementary isotropic subspaces V, V ⊂ so(n, n).…”

“…For pseudo-Riemannian manifolds of signature different from Riemannian and Lorentzian ones, only some partial case are considered [6,7,10,11,15,16,20,23]. There is also a classification of connected irreducible holonomy groups of torsion-free affine connections [25]; the groups corresponding to the Ricci-flat case are found in [4].…”

“…Later this classification was corrected [1], and it was proved that all the obtained algebras can be realized as the holonomy algebras of pseudo-Riemannian manifolds [12,22]. Here is the list of these algebras: so(p, q), so(p, C) ⊂ so(p, p), u(r, s) ⊂ so(2r, 2s), su(r, s) ⊂ so(2r, 2s), sp(r, s) ⊂ so(4r, 4s), sp(r, s) ⊕ sp(1) ⊂ so(4r, 4s), (4,4), spin(7, C) ⊂ so (8,8), 7). Consider now the Einstein condition.…”

Holonomy groups of Lorentzian manifolds

“…To that in the paper [4] the holonomy groups of the cones over pseudo-Riemannian manifolds, and in particular over Lorentzian manifolds, are studied. There are results about irreducible holonomy groups of linear torsion-free connections [9], [39], [104], [111]. The holonomy groups are defined also for manifolds with conformal metrics,in particular, these groups allow to decide if there are Einstein metrics in the conformal class [14].…”