In differential geometry, a pseudo-Riemannian manifold  (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the metric tensor need not be positive-definite. Instead a weaker condition of nondegeneracy is imposed.
In differential geometry, a differentiable manifold is a space which 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 which can be mapped into n-dimensional Euclidean space.
Tangent spaces and metric tensors
Associated with each point in an -dimensional differentiable manifold is a tangent space (denoted ). This is an -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 which assigns a real number to pairs of tangent vectors at each tangent space of the manifold. Denoting the metric tensor by we can express this as
The map is symmetric and bilinear so if are tangent vectors at a point to the manifold then we have
for any real number .
That is non-degenerate means there are no non-zero such that for all .
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 rigidity theorem, 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. For a non-degenerate metric tensor r = 0 and the signature may be denoted (p,q), where p + q = n.
A pseudo-Riemannian manifold is a differentiable manifold equipped with a non-degenerate, smooth, symmetric metric tensor which, unlike a Riemannian metric, need not be positive-definite, but must be non-degenerate. Such a metric is called a pseudo-Riemannian metric and its values can be positive, negative or zero.
The signature of a pseudo-Riemannian metric is (p, q) where both p and q are non-negative.
A Lorentzian manifold is an important special case of a pseudo-Riemannian manifold in which the signature of the metric is (1, n−1) (or sometimes (n−1, 1), see sign convention). Such metrics are called Lorentzian metrics. They are named after the physicist Hendrik Lorentz.
Applications in physics
After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important because of their physical applications to the theory of general relativity.
A principal basis 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, a signature of (p, 1) or (1, q) allows tangent vectors to be classified into timelike, null or spacelike (see Causal structure).
Properties of pseudo-Riemannian manifolds
Just as Euclidean space can be thought of as the model Riemannian manifold, Minkowski space with the flat Minkowski metric is the model Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature (p, q) is with the metric
Some basic theorems of Riemannian geometry can be generalized to the pseudo-Riemannian case. In particular, the fundamental theorem of Riemannian geometry is true of pseudo-Riemannian manifolds as well. 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 which 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. The Clifton–Pohl torus provides an example of a pseudo-Riemannian manifold that is compact but not complete, a combination of properties that the Hopf–Rinow theorem disallows for Riemannian manifolds.
- Hyperbolic partial differential equation
- Causality conditions
- Globally hyperbolic manifold
- Benn, I.M.; Tucker, R.W. (1987), An introduction to Spinors and Geometry with Applications in Physics, Adam Hilger, p. 172
- Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds, The Macmillan Company, p. 208
- O'Neill, Barrett (1983), Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics 103, Academic Press, p. 193, ISBN 9780080570570.
- Benn, I.M.; Tucker, R.W. (1987), An introduction to Spinors and Geometry with Applications in Physics (First published 1987 ed.), Adam Hilger, ISBN 0-85274-169-3
- Bishop, Richard L.; Goldberg, Samuel I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6
- Chen, Bang-Yen (2011), Pseudo-Riemannian Geometry, [delta]-invariants and Applications, World Scientific Publisher, ISBN 978-981-4329-63-7
- G. Vrănceanu & R. Roşca (1976) Introduction to Relativity and Pseudo-Riemannian Geometry, Bucarest: Editura Academiei Republicii Socialiste România.