Consistent Kaluza-Klein sphere reductions

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 IIB supergravity), the only other possibilities that allow consistent reductions are for $p=2,$ reduced on S², and for $p=3,$ reduced on S³ or S^D-3. We construct the fully non-linear Kaluza-Klein Ans\"atze 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 $D=10.$