  There are various ways of defining the Wick rotation in a gravitational
context. There are good arguments to view it as an analytic continuation of the
metric, instead of the coordinates. We focus on one very general definition and
argue that it is incompatible with the requirement of preserving the field
equations and the symmetries at global level: in some cases the Euclidean
metric cannot be defined on the original Lorentzian manifold but only on a
submanifold. This phenomenon is related to the existence of horizons, as
illustrated in the cases of the de Sitter and Schwarzschild metrics.
