The metric tensor on a smooth manifold is said to be semi-Riemannian if the index of is nonzero.
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
with the rightmost summand nonzero (Snygg 2012).