  We prove a non-existence theorem for smooth, supersymmetric, warped AdS6
solutions with connected, compact without boundary internal space in D=11 and
(massive) IIA supergravities. In IIB supergravity we show that if such AdS6
solutions exist, then the NSNS and RR 3-form fluxes must be linearly
independent and certain spinor bi-linears must be appropriately restricted.
Moreover we demonstrate that the internal space admits an so(3) action which
leaves all the fields invariant and for smooth solutions the principal orbits
must have co-dimension two. We also describe the topology and geometry of
internal spaces that admit such a so(3) action and show that there are no
solutions for which the internal space has topology F * S^2, where F is an
oriented surface.
