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?



Obtain Lorentzian manifolds from Riemannian ones by Wick rotation
This is a somewhat different take on Igor's answer, and I offer it just in case you are interested.


First, one doesn't need to have any continuous symmetries in order to have this kind of 'Wick rotation' exist. For example, if $(M,g)$ is a real-analytic Riemannian manifold that admits a nontrivial isometric involution $\iota:M\to M$ that fixes a hypersurface $H\subset M$, then, near $H$, one can write the metric in the form $g = dt^2 + h(t^2)$ where $t$ is the distance from the hypersurface and $h(a)$ is the induced metric on the level sets $t^2 = a$. Then the Wick rotation is just $g' = -d\tau^2 + h(-\tau^2)$ in the sense that this is the Lorentian metric induced on the slice $t = i\tau$ in the complexification $(M^\mathbb{C},g^\mathbb{C})$.


Second, one could generalize things a bit and say that two real-analytic (pseudo-)Riemannian metrics are 'Wick-related' if they are '$\mathbb{R}$-slices' of a common connected holomorphic Riemannian complex $n$-manifold $(M^\mathbb{C},g^\mathbb{C})$. By an '$\mathbb{R}$-slice', I mean a real $n$-manifold $N\subset M^\mathbb{C}$ such that the pullback of $g^\mathbb{C}$ to $N$ is real-valued and nondegenerate. In this terminology, I think that two $\mathbb{R}$-slices $N\_1,N\_2\subset M^\mathbb{C}$ should be said to be related by a 'Wick rotation' if $N\_1\cap N\_2$ is a submanifold of dimension $n{-}1$. This is certainly what happens in the case above generated by an isometric involution fixing a hypersurface. (*Added note:* In fact, I don't know an example of a Wick-rotation in this sense that isn't generated by such an isometric involution. It might be interesting to try to prove that this does give them all or else find a counterexample. I note that, to second order, it is true: The hypersurface $N\_1\cap N\_2$ is always totally geodesic in each of $N\_1$ and $N\_2$ (with their induced metrics), so reflection in the hypersurface is an isometry at least up to second order.)


Then the problem becomes how to tell when a given connected holomorphic Riemannian complex $n$-manifold $(M^\mathbb{C},g^\mathbb{C})$ has an $\mathbb{R}$-slice (and, of course, to determine them when they exist). Obviously, the complexification of a real-analytic pseudo-Riemannian $n$-manifold has at least one $\mathbb{R}$-slice, but generically, when $n>1$, this is the only one, so 'most' real-analytic pseudo-Riemannian manifolds are not Wick-related to any other, let alone possess a Wick-rotation.


The reason is that $\mathbb{R}$-slices are the integral manifolds of a very restrictive system of PDE for real submanifolds of $(M^\mathbb{C},g^\mathbb{C})$: If one lets $R\subset \mathrm{Gr}^\mathbb{R}\_n(TM^\mathbb{C})$ denote the set of real $n$-planes $E\subset T\_pM^\mathbb{C}$ to which $g^\mathbb{C}$ restricts to be real-valued (and, of course, nondegenerate), then $R$ is a smooth manifold of dimension $2n+\tfrac12n(n{-}1)$ with $n{+}1$ components (one for each possible index of $g$ when restricted to the $n$-plane $E$), and the basepoint projection $\pi:R\to M^\mathbb{C}$ given by $\pi(E) = p$ is a smooth submersion. It is easy to show that, for any given $E\in R$, there is at most one $\mathbb{R}$-slice that has $E$ as a tangent space. This is because there is a canonical $n$-plane field $H$ on $R$ with the property that the set of tangent spaces of any $\mathbb{R}$-slice is an $n$-manifold in $R$ that is tangent to $H$ everywhere.


The only time $H$ is Frobenius is when $(M^\mathbb{C},g^\mathbb{C})$ is the complexification of a (real) space form, i.e., a Riemannian manifold of constant sectional curvature. Generically, $H$ has no integral manifolds at all, and, generically, when it does have one, it has only one connected component.


For example, in the case $n=2$, if the Gauss curvature $K$ of $(M^\mathbb{C},g^\mathbb{C})$ is not constant, then it has at most a $1$-parameter family of positive definite $\mathbb{R}$-slices, and this happens only when the $\mathbb{R}$-slices are all isometric and have a symmetry vector field (so they are locally surfaces of revolution). 


Now, in fact, each $\mathbb{R}$-slice (of whatever index) lies in the (real) hypersurface in $M^\mathbb{C}$ on which $K$ takes values in $\mathbb{R}$. Set $E = g^\mathbb{C}(\nabla K,\nabla K)$ (everything computed in the holomorphic category). For most metrics $g^\mathbb{C}$, the holomorphic functions $K$ and $E$ will not be functionally dependent, i.e., $\mathrm{d}K\wedge\mathrm{d}E$ will vanish only on a (possibly empty) complex-analytic subvariety $C\subset M^\mathbb{C}$ of complex dimension $1$. Go ahead and define $F = g^\mathbb{C}(\nabla K,\nabla E)$ and $G = g^\mathbb{C}(\nabla E,\nabla E)$. 


Obviously, any $\mathbb{R}$-slice $N\subset M^\mathbb{C}$ must lie inside the locus on which the holomorphic functions $K$, $E$, $F$, and $G$ assume real values. Outside of $C$, the set $L\subset M^\mathbb{C}\setminus C$ on which $K$ and $E$ take real values is a (possibly empty) real submanifold of dimension $2$ and its components are the only possible $\mathbb{R}$-slices that lie outside of $C$ (of course, any $\mathbb{R}$-slice can only intersect $C$ in a (real) $1$-dimensional curve at most). A component of $L$ actually is an $\mathbb{R}$-slice if and only if $F$ and $G$ are real-valued on it. Two components are related by a Wick-rotation in the above sense if they intersect (in a real $1$-dimensional curve that necessarily lies inside $C$).


A similar, but slightly more involved analysis can be done for the case in which $K$ and $E$ are dependent everywhere on $M^\mathbb{C}$. Moreover, a similar, but more complicated analysis can be used in higher dimensions, with, say, the symmetric functions of the eigenvalues of the holomorphic Ricci tensor used in the place of $K$.


