Given a Lorentzian spacetime (M, g) and a non-vanishing timelike vector field u(γ) with level surfaces σ, one can construct on M a Euclidean metric g-1E= g-1+ 2u ⊗u (Hawking and Ellis 1973 The Large Scale Structure of Space-Time (Cambridge: Cambridge University Press)). Motivated by this, we consider a class of metrics ∧g-1 = g-1-⊖ (γ) u⊗u with an arbitrary function ⊖that interpolates between the Euclidean ⊖= -2) and Lorentzian (⊖= 0) regimes, separated by the codimension one hypersurface ∑0 defined by ⊖= -1. Since ∧g can not, in general, be obtained from g by a diffeomorphism, its Euclidean regime is in general different from that obtained from Wick rotation t →-it. For example, if g is the k = 0 Lorentzian de Sitter metric corresponding to Λ> 0, the Euclidean regime of ∧g is the k = 0 Euclidean anti-de Sitter space with Λ < 0. We analyze the curvature tensors associated with ∧g for arbitrary Lorentzian metrics g and timelike geodesic fields u, and show that they have interesting and remarkable mathematical structures: (i) Additional terms arise in the Euclidean regime ⊖→-2 of ∧g g. (ii) For the simplest choice of a step-profile for ⊖, the Ricci scalar Ric[∧g ] of ∧g reduces, in the Lorentzian regime ⊖→ 0, to the complete EinsteinHilbert lagrangian with the correct GibbonsHawkingYork boundary term; the latter arises as a delta-function of strength 2K supported on sum;0. (iii) In the Euclidean regime ⊖→-2, Ric[∧g] also has an extra term 23R of the u-foliation. We highlight similar foliation dependent terms in the full Riemann tensor. We present some explicit examples for FLRW spacetimes in standard foliation and spherically symmetric spacetimes in the Painleve-Gullstrand foliation. We briefly discuss implications of the results for Euclidean quantum gravity and quantum cosmology.