TITLE: Obtain Lorentzian manifolds from Riemannian ones by Wick rotation
QUESTION [20 upvotes]: In some cases, Wick rotation of a metric, formally consisting in substituting a coordinate with i times the coordinate itself, allows one to construct a Riemannian manifold starting from a Lorentzian one. The most known example, is Minkowski spacetime, which becomes Euclidean 4-space. Other example are Schwarzschild and Taub-NUT.
I would be interested in knowing which conditions are needed to ensure that by performing a Wick rotation on a Lorentzian manifold one obtains a Riemannian manifold and what problems can arise in the process - i.e. why one cannot always obtain a Riemannian manifold by Wick rotating a Lorentzian one. I am also interested in the reverse process, i.e. under which conditions one can obtain a Lorentzian manifold starting from a Riemannian one.
Are there any properties of the manifold which are preserved by the process?
EDIT: Thanks for the comments and answers so far. Indeed, I do not have a definition of what is meant by a Wick rotation for general manifolds - perhaps I should have asked how and if can such a notion be defined in a general context.
As far as I know twistors theory makes use of complexified manifolds, is anyone aware of whether such a theory has anything to say about the problem?

REPLY [2 votes]: See also the recent paper  
A Wick-rotatable metric is purely electric by C. Helleland and S. Hervik, arXiv:1504.01244