Ricci-flat holonomy: a classification (Q886336)

Irreducible holonomy algebras \(\mathfrak{h}\) of torsion-free connections were classified by \textit{S. Merkulov} and \textit{L. Schwachhöfer} [Ann. Math. (2) 150, No. 1, 77--149 (1999); addendum ibid. 150, No. 3, 1177--1179 (1999; Zbl 0992.53038)]. Given such an algebra \(\mathfrak{h}\), one can distinguish three cases: (i) Any torsion-free connection with holonomy \(\mathfrak{h}\) is Ricci-flat. (ii) Any torsion-free connection with holonomy \(\mathfrak{h}\) has non-zero Ricci curvature. (iii) There exist torsion-free connections with holonomy \(\mathfrak{h}\) which are Ricci-flat and others which are not Ricci-flat. The author classifies the irreducible holonomy algebras according to these cases. This is part of his PhD thesis at Oxford University.