  The geometric aspect of Wick rotation in quantum field theory and its
localization on manifolds are explored. After the explanation of the notion and
its related geometric objects, we study the topology of the set of landing $W$
for Wick rotations and its natural stratification. These structures in two,
three, and four dimensions are computed explicitly. We then focus on more
details in two dimensions. In particular, we study the embedding of $W$ in the
ambient space of Wick rotations, the resolution of the generic metric
singularities of a Lorentzian surface $\Sigma$ by local Wick rotations, and
some related $S^1$-bundles over $\Sigma$.
