A pseudo-Riemannian manifold is a smooth manifold equipped with a pseudo-Riemannian metric,
that is, a symmetric nondegenerate covariant tensor field of type 
with constant index.
It is also called a semi-Riemannian manifold
(Burtscher 2022, Definition 2.22).
Under this convention, the index may be any integer from 0 to the dimension of the manifold. Riemannian manifolds have index 0, and the case with one negative square and all remaining squares positive is a Lorentzian manifold when the dimension is at least 2. Some authors reserve the term pseudo-Riemannian for metrics that are not positive definite.
