  We discuss the generic geometric properties of metrics $\widehat {g}_{ab}$
constructed from Lorentzian metric $g_{ab}$ and a nowhere vanishing,
hypersurface orthogonal, timelike vector field $u^a$. The metric ${\widehat
g}_{ab}$ has Euclidean signature in a certain domain, with the transition to
Lorentzian signature occurring at some hypersurface $\Sigma$ orthogonal to
$u^a$. Geometry associated with ${\widehat g}_{ab}$ has recently been shown to
yield remarkable new insights for classical and quantum gravity. In this work,
we prove several general results applicable in physically relevant spacetimes
for congruences $u^i$ with non-zero acceleration $a^i$. We present as examples
the cases of dynamical spherically symmetric spacetimes and spacetimes with
maximal symmetry. We also investigate this formalism within the context of
thermal effects in curved spacetimes with horizons. Specifically, we discuss:
(i) the Holonomy of loops lying partially or wholly in the Euclidean regime. We
show that the contribution of the Euclidean domain to holonomy is completely
determined by extrinsic curvature $K_{ab}$ of $\Sigma$ and acceleration $a^i$.
(ii) We also compute entropy using this formalism for simple field theories and
obtain foliation dependent corrections for the Lanczos-Lovelock gravity,
Bekenstein-Hawking entropy relation in four spacetime dimensions.
