Open main menu

Diffgeom β

Lorentzian manifold

This article defines a differential manifold with the following additional structure -- the structure group is reduced to: Lorentzian group viz a group of the form O(n,1)


A Lorentzian manifold is the following data:

  • A differential manifold M
  • A symmetric nondegenerate bilinear form of type (n,1) on each tangent space, such that the form varies smoothly with the point.

In other words, a Lorentzian manifold is the reduction of the structure group of a differential manifold to the Lorentzian group O(n,1).

Lorentzian manifolds are special cases of pseudo-Riemannian manifolds. Thus, all the generic constructs for a pseudo-Riemannian manifold apply to Lorentzian manifolds.

Relation with other structures

Weaker structures