On imaginary time I've heard a few times that "the time was imaginary before the Big Bang".
I am guessing it means that at this stage, the space-time was a Riemannian $4$-manifold, but I am not sure this guess is correct.
If this is right, then I would like to know more about this Riemannian manifold.
But where a mathematician can learn about it?
Is there by any chance a book adapted to geometers? (otherwise it will be very hard to read.)
 A: The introduction of imaginary time as a way to resolve the Big Bang singularity is a proposal by Hawking and others. I don't think it plays a role in modern cosmology, see Emerging from imaginary time. The main difficulty is how to join the 4-dimensional region with Euclidean metric to a 3+1 dimensional space-time region with a Lorentzian metric.
The modern understanding is that time is emergent, not "imaginary". Loop-quantum-gravity is one theory that attempts to formalize this. It is not formulated starting from a metric on space-time, the very existence of three space dimensions and one time dimension is derived from a more fundamental framework.
Thinking of time as an emergent property is a bit like how we think of temperature. The fundamental theory of gas at the molecular level does not have temperature as a variable. This variable arises ("emerges") when one tries to describe the statistical properties of many molecules. In this sense time did not exist at the Big Bang singularity. It emerged as the universe developed.
