  We obtain a large class of AdS spacetimes warped with certain internal spaces
in eleven-dimensional and type IIA/IIB supergravities. The warp factors depend
only on the internal coordinates. These solutions arise as the near-horizon
geometries of more general semi-localised multi-intersections of $p$-branes. We
achieve this by noting that any sphere (or AdS spacetime) of dimension greater
than 3 can be viewed as a foliation involving S^3 (or AdS_3). Then the S^3 (or
AdS_3) can be replaced by a three-dimensional lens space (or a BTZ black hole),
which arises naturally from the introduction of a NUT (or a pp-wave) to the
M-branes or the D3-brane. We then obtain multi-intersections by performing a
Kaluza-Klein reduction or Hopf T-duality transformation on the fibre coordinate
of the lens space (or the BTZ black hole). These geometries provide further
possible examples of the AdS/CFT correspondence and of consistent embeddings of
lower-dimensional gauged supergravities in D=11 or D=10.
