In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.

Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space.

A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.

In differential geometry, a differentiable manifold is a space that is locally similar to a Euclidean space. In an n-dimensional Euclidean space any point can be specified by n real numbers. These are called the coordinates of the point.

An n-dimensional differentiable manifold is a generalisation of n-dimensional Euclidean space. In a manifold it may only be possible to define coordinates locally. This is achieved by defining coordinate patches: subsets of the manifold that can be mapped into n-dimensional Euclidean space.

See Manifold, Differentiable manifold, Coordinate patch for more details.

Associated with each point ${\displaystyle p}$ in an ${\displaystyle n}$-dimensional differentiable manifold ${\displaystyle M}$ is a tangent space (denoted ${\displaystyle T_{p}M}$). This is an ${\displaystyle n}$-dimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point ${\displaystyle p}$.

A metric tensor is a non-degenerate, smooth, symmetric, bilinear map that assigns a real number to pairs of tangent vectors at each tangent space of the manifold. Denoting the metric tensor by ${\displaystyle g}$ we can express this as