Busemann's metric theory of timelike spaces

International audienceBusemann's theory of timelike spaces is a theory of semi-Riemannian manifolds -- a generalization of Riemannian manifolds where the quadratic form defining the metric is not required to be positive definite -- from the metrical point of view, that is, without any use of tangent spaces and differentiability. This theory is a generalization of Lorentzian geometry, a semi-Riemannian geometry which is associated to a quadratic form of signature $(-,+,..., +)$. Such a geometry underlines the theory of relativity. As a matter of fact, the term ``timelike" used by Busemann refers to the physical theory of relativity, where space and time are combined into a single space, the so-called Minkowski or Lorentz spacetime. The first work concerning the geometrical setting of special relativity was done by Hermann Minkowski in the early 1900s. According to this theory, the notions of space and time are themselves relative: a point in space does not exist in the absolute, it depends on time, and conversely, time depends on the observer. Only space-time exists. In the axioms of timelike spaces, one starts with a topological space equipped with two basic objects: a distance function, which plays the role of a (degenerate, or indefinite) metric, and a partial order relation. This order relation corresponds to the causality property of the space-time of relativity theory. Busemann's metric theory is based on a list of axioms that concern the order relation, the distance function and their relation to each other. There is also a local theory. Busemann gave applications of his theory to the structure of projective transformations of convex sets.We review Busemann's theory. We shall describe the main axioms, and then state the main results and the main questions that arise from that theory. We also explain the motivation and the relation with relativity theory. We also review the work on this subject that was done by Busemann's students