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 $\mathrm{SO}(n+1\mathrm{}),$ 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 $\mathrm{}(D,p)=(11,4)\mathrm{}$ and (10,5) (associated with the $S^4$ and $S^7$ reductions of $D=11\mathrm{}$ supergravity, and the $S^5$ reduction of type IIB supergravity), the only other possibilities that allow consistent reductions are for $p=2,$ reduced on $S^2,$ and for $p=3,$ reduced on $S^3$ or $S^{D-3}.$ We construct the fully non-linear Kaluza-Klein Ansätze in all these cases. In particular, we obtain $D=3,$ $N=8,$ $\mathrm{SO}\left(8\right)\mathrm{}$ and $D=7,$ $N=2,$ $\mathrm{SO}\left(4\right)\mathrm{}$ gauged supergravities from $S^7$ and $S^3$ reductions of $N=1\mathrm{}$ supergravity in $D=10.\mathrm{}$