Ricci-flat holonomy: A classification

Ricci-Flat Holonomy: a Classification

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...

Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebr...

Ricci-flat supertwistor spaces

We show that supertwistor spaces constructed as a Kähler quotient of a hyperkähler cone (HKC) with equal numbers of bosonic and fermionic coordinates are Ricci-flat, and hence, Calabi-Yau. We study deformations of the supertwistor space induced from deformations of the HKC. We also discuss general infinitesimal deformations that preserve Ricci-flatness.

On Ricci flat supermanifolds

We study the Ricci flatness condition on generic supermanifolds. It has been found recently that when the fermionic complex dimension of the supermanifold is one the vanishing of the super-Ricci curvature implies the bosonic submanifold has vanishing scalar curvature. We prove that this phenomena is only restricted to fermionic complex dimension one. Further we conjecture that for complex fermi...

Conformal Holonomy: a Classification