pseudoriemannian

Pseudo-Riemannian geometry studies differentiable manifolds equipped with a smooth, nondegenerate, symmetric bilinear form on each tangent space, called a pseudo-Riemannian metric. Unlike Riemannian metrics, these are not required to be positive definite; they have a fixed signature (p, q) with p + q = n, the dimension of the manifold. When p = 0 or q = 0, the metric is Riemannian; when p = 1 and q = n − 1, the metric is Lorentzian, which is central to spacetime models in physics.

A pseudo-Riemannian metric g enables the Levi-Civita connection, the unique torsion-free, metric-compatible connection. This yields notions of parallel transport, covariant differentiation, and geodesics—curves that locally extremize the metric’s interval. The Riemann curvature tensor, derived from the connection, encodes intrinsic curvature. The indefinite signature induces a causal structure in Lorentzian and related settings, distinguishing timelike, spacelike, and null vectors.

Distinctions from the Riemannian case include the behavior of distances: the metric does not generally define a globally well-behaved distance function, and length notions for curves must be interpreted with attention to sign and causal type. Nevertheless, many concepts from Riemannian geometry extend, such as curvature, geodesic completeness, and global analysis, and the framework underpins significant physical theories.

Examples include Minkowski space (flat Lorentzian, signature (1, n−1)) and curved spacetimes such as de Sitter and anti-de Sitter spaces. Abstractly, a pseudo-Riemannian manifold is a pair (M, g) with g a smooth, nondegenerate section of Sym^2(T*M), and the local isometry group is O(p, q).