# 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

## Definition

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.