A pseudo-Riemannian manifold is a generalization of a Riemannian manifold that allows for the metric tensor to have signature that is not positive definite. While in a Riemannian manifold the metric tensor \( g \) is positive definite, which means that for any nonzero tangent vector \( v \), the inner product \( g(v, v) > 0 \), a pseudo-Riemannian manifold has a metric tensor that can have both positive and negative eigenvalues.
Articles by others on the same topic
A pseudo-Riemannian manifold is a smooth manifold with a smooth metric tensor that is a nondegenerate bilinear form on each tangent space. Positivity is not required. A Riemannian manifold is the positive-definite case, while a Lorentzian manifold has exactly one temporal direction in its metric signature.