The metric tensor g on a smooth manifold M = (M, g) is said to be semi-Riemannian if the index of g 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 g fails to be positive definite. Alternatively, a manifold is semi-Riemannian (or pseudo-Riemannian) if its infinitesimal distance (d s)^2 is equivalent to that of a pseudo-Euclidean space of signature (p, q) for q!=0, i.e., if (d s)^2 = sum_(j = 1)^p (d x^j)^2 - sum_(j = p + 1)^n (d x^j)^2 with the rightmost summand nonzero.

Lorentzian manifold | metric signature | metric tensor | metric tensor index | positive definite tensor | pseudo-Euclidean space | pseudo-Riemannian manifold | semi-Riemannian manifold | smooth manifold | strong pseudo-Riemannian metric | strong Riemannian metric | weak pseudo-Riemannian metric | weak Riemannian metric

Bernhard Riemann