A Lorentzian manifold is the following data:
- A differential manifold
- A symmetric nondegenerate bilinear form of type 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 .
Lorentzian manifolds are special cases of pseudo-Riemannian manifolds. Thus, all the generic constructs for a pseudo-Riemannian manifold apply to Lorentzian manifolds.