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.
This is achieved by defining coordinate patches: subsets of the manifold that can be mapped into n-dimensional Euclidean space.
-dimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point
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.
Given a metric tensor g on an n-dimensional real manifold, the quadratic form q(x) = g(x, x) associated with the metric tensor applied to each vector of any orthogonal basis produces n real values.
By Sylvester's law of inertia, the number of each positive, negative and zero values produced in this manner are invariants of the metric tensor, independent of the choice of orthogonal basis.
The signature (p, q, r) of the metric tensor gives these numbers, shown in the same order.
The non-degeneracy condition together with continuity implies that p and q remain unchanged throughout the manifold (assuming it is connected).
A principal premise of general relativity is that spacetime can be modeled as a 4-dimensional Lorentzian manifold of signature (3, 1) or, equivalently, (1, 3).
Unlike Riemannian manifolds with positive-definite metrics, an indefinite signature allows tangent vectors to be classified into timelike, null or spacelike.
With a signature of (p, 1) or (1, q), the manifold is also locally (and possibly globally) time-orientable (see Causal structure).
can be thought of as the local model of a Riemannian manifold, Minkowski space
with the flat Minkowski metric is the local model of a Lorentzian manifold.
, for which there exist coordinates xi such that Some theorems of Riemannian geometry can be generalized to the pseudo-Riemannian case.
In particular, the fundamental theorem of Riemannian geometry is true of all pseudo-Riemannian manifolds.
This allows one to speak of the Levi-Civita connection on a pseudo-Riemannian manifold along with the associated curvature tensor.
On the other hand, there are many theorems in Riemannian geometry that do not hold in the generalized case.
For example, it is not true that every smooth manifold admits a pseudo-Riemannian metric of a given signature; there are certain topological obstructions.
Furthermore, a submanifold does not always inherit the structure of a pseudo-Riemannian manifold; for example, the metric tensor becomes zero on any light-like curve.