Pseudo-Riemannian manifold

This article defines a differential manifold with the following additional structure -- the structure group is reduced to: pseudo-orthogonal group

Definition

A pseudo-Riemannian manifold or semi-Riemannian manifold is the following data:

• A differential manifold $M$
• A symmetric nondegenerate bilinear form of type $(p,q)$ for fixed integers $p$ and $q$, on the tangent space at each point, such that the form varies smoothly with the point.

In other words, a pseudo-Riemannian manifold is a reduction of the structure group to $O(p,q)$ on a differential manifold.