In the usual procedure for toroidal Kaluza-Klein reduction, all the
higher-dimensional fields are taken to be independent of the coordinates on the
internal space. It has recently been observed that a generalisation of this
procedure is possible, which gives rise to lower-dimensional ``massive''
supergravities. The generalised reduction involves allowing gauge potentials in
the higher dimension to have an additional linear dependence on the toroidal
coordinates. In this paper, we show that a much wider class of generalised
reductions is possible, in which higher-dimensional potentials have additional
terms involving differential forms on the internal manifold whose exterior
derivatives yield representatives of certain of its cohomology classes. We
consider various examples, including the generalised reduction of M-theory and
type II strings on K3, Calabi-Yau and 7-dimensional Joyce manifolds. The
resulting massive supergravities support domain-wall solutions that arise by
the vertical dimensional reduction of higher-dimensional solitonic p-branes and
intersecting p-branes.Comment: Latex, 24 pages, no figures, typo corrected, reference added and
discussion of duality extende
