A pseudo-Riemannian metric on a finite-dimensional smooth manifold is a smooth symmetric nondegenerate covariant tensor field of type 
. Its index is constant
on each connected component. It is also called
a semi-Riemannian metric (Burtscher 2022,
Definition 2.22).
Under this convention, a Riemannian metric is a pseudo-Riemannian metric of index 0. Some authors restrict the term pseudo-Riemannian to metrics that are not positive definite. The Minkowski metric is the flat metric on the Lorentzian manifold Minkowski space, a particular example of a pseudo-Riemannian metric.
