  Gravitational instantons, solutions to the euclidean Einstein equations, with
topology $R^3 XS^1$ arise naturally in any discussion of finite temperature
quantum gravity. This Letter shows that all such instantons (irrespective of
their interior behaviour) must have the same asymptotic structure as the
Schwarzschild instanton. Using this, it can be shown that if the Ricci tensor
of the manifold is non-negative it must be flat. One special case is when the
Ricci tensor vanishes; hence one can conclude that there is no nontrivial
vacuum gravitational instanton. This result has uses both in quantum and
classical gravity. It places a significant restriction on the instabilities of
hot flat space. It also can be used to show that any static vacuum Lorentzian
Kaluza-Klein solution is flat.
