We study the circumstances under which a Kaluza-Klein reduction on an n-sphere, with a massless truncation that includes all the Yang-Mills fields of SO(n+1), can be consistent at the full non-linear level. We take as the starting point a theory comprising a p-form field strength and (possibly) a dilaton, coupled to gravity in the higher dimension D. We show that aside from the previously studied cases with (D,p)=(11,4) and (10,5) (associated with the S4 and S7 reductions of D=11 supergravity, and the S5 reduction of type HB supergravity), the only other possibilities that allow consistent reductions are for p=2, reduced on S2, and for p=3, reduced on S3 or SD-3. We construct the fully non-linear Kaluza-Klein Anstze in all these cases. In particular, we obtain D=3, N=8, SO(8) and D=7, N=2, SO(4) gauged supergravities from S7 and S3 reductions of N=1 supergravity in D=10. 2000 The American Physical Society.
