Manifolds Obtained by Soldering Together Points, Lines, Etc

where Rij is the Ricci curvature tensor of the metric and R is its scalar curvature. This equation tells us how the metric tensor gij (i.e. the field mediating the gravitation) is obeying to constraints imposed by the energy-momentum tensor Tij , which encodes informations about the distribution of energy and momentum in spacetime. For simplicity we will assume in the following that Tij = 0, i.e. we consider the Einstein equation in the vacuum. Note that if we work with equation (1) in the same way as for any other partial differential equation, then it is very easy to forget about the original intuition of Einstein (which is that the space-time is built out of the gravitational field gij) and erroneously to implicitely assume that we are given some manifold M, and we are looking for an unknown field gij satisfying (1). We want to take this physical intuition seriously into account and to look for some mathematical framework which would help us to keep in mind that the manifold M and the metric should be built simultaneously when solving equation (1). From this point of view the only ‘kinematic’ condition which is imposed is that, at each point of the spacetime, the tangent space to it is endowed with a metric (which is a Minkowski metric in the physical case of pseudo-Riemannian manifolds and an Euclidean one in the Riemannian