  We investigate connections between pairs of (pseudo-)Riemannian metrics whose
sum is a (tensor) product of a covector field with itself. A bijective mapping
between the classes of Euclidean and Lorentzian metrics is constructed as a
special result. The existence of such maps on a differentiable manifold is
discussed. Similar relations for metrics of arbitrary signature on a manifold
are considered. We point the possibility that any physical theory based on real
Lorentzian metric(s) can be (re)formulated equivalently in terms of real
Euclidean metric(s).
