The aim of this paper is two-fold. First, we provide a simple and pedagogical discussion of how compactifications of M-theory or supergravity preserving some four-dimensional supersymmetry naturally lead to reduced holonomy or its generalization, reduced weak holonomy. We relate the existence of a (conformal) Killing spinor to the existence of certain closed and co-closed p-forms, and to the metric being Ricci flat or Einstein. Then, for seven-dimensional manifolds, we show that octonionic self-duality conditions on the spin connection are equivalent to G(2) holonomy and certain generalized self-duality conditions to weak G(2) holonomy. The latter lift to self-duality conditions for cohomogeneity-one spin(7) metrics. To illustrate the power of this approach. we present several examples where the self-duality condition largely simplifies the derivation of a G(2) or weak G(2) metric. (C) 2002 Published by Elsevier Science B.V. [References: 35]
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