A smooth manifold is said to be semi-Riemannian if the indexMetric
Tensor Index of is nonzero. Alternatively, a smooth manifold is semi-Riemannian
provided that it comes equipped with a semi-Riemannian
metric.

In nearly all literature, the term semi-Riemannian is used synonymously with the term pseudo-Riemannian and is used
to describe manifolds whose metric tensor fails to be positive
definite. Alternatively, a manifold is semi-Riemannian (or pseudo-Riemannian)
if its infinitesimal distance is equivalent to that of a pseudo-Euclidean
space of signature for , i.e., if