  We apply duality rotations and complex transformations to the Schwarzschild
metric to obtain wormhole geometries with two asymptotically flat regions
connected by a throat. In the simplest case these are the well-known wormholes
supported by phantom scalar field. Further duality rotations remove the scalar
field to yield less well known vacuum metrics of the oblate Zipoy-Voorhees-Weyl
class, which describe ring wormholes. The ring encircles the wormhole throat
and can have any radius, whereas its tension is always negative and should be
less than $-c^4/4G$. If the tension reaches the maximal value, the geometry
becomes exactly flat, but the topology remains non-trivial and corresponds to
two copies of Minkowski space glued together along the disk encircled by the
ring. The geodesics are straight lines, and those which traverse the ring get
to the other universe. The ring therefore literally produces a hole in space.
Such wormholes could perhaps be created by negative energies concentrated in
toroidal volumes, for example by vacuum fluctuations.
