Abstract
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 × S2, where F is an oriented surface.