Inmathematical physics, apseudo-Riemannian manifold,^[1][2] also called asemi-Riemannian manifold, is adifferentiable manifold with ametric tensor that is everywherenondegenerate. This is a generalization of aRiemannian manifold in which the requirement ofpositive-definiteness is relaxed.

Everytangent space of a pseudo-Riemannian manifold is apseudo-Euclidean vector space.

A special case used ingeneral relativity is a four-dimensionalLorentzian manifold for modelingspacetime, where tangent vectors can be classified astimelike, null, and spacelike.

Indifferential geometry, adifferentiable manifold is a space that is locally similar to aEuclidean space. In ann-dimensional Euclidean space any point can be specified byn real numbers. These are called thecoordinates of the point.

Ann-dimensional differentiable manifold is a generalisation ofn-dimensional Euclidean space. In a manifold it may only be possible to define coordinateslocally. This is achieved by definingcoordinate patches: subsets of the manifold that can be mapped inton-dimensional Euclidean space.

SeeManifold,Differentiable manifold,Coordinate patch for more details.

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

Ametric tensor is anon-degenerate, smooth, symmetric,bilinear map that assigns areal 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

${\displaystyle g:T_{p}M\times T_{p}M\to \mathbb{R}.}$ 

The map is symmetric and bilinear so if${\displaystyle X,Y,Z\in T_{p}M}$  are tangent vectors at a point${\displaystyle p}$  to the manifold${\displaystyle M}$  then we have

• ${\displaystyle g(X,Y)=g(Y,X)}$ 
• ${\displaystyle g(aX+Y,Z)=ag(X,Z)+g(Y,Z)}$ 

for any real number${\displaystyle a\in \mathbb{R}}$ .

That${\displaystyle g}$  isnon-degenerate means there is no non-zero${\displaystyle X\in T_{p}M}$  such that${\displaystyle g(X,Y)=0}$  for all${\displaystyle Y\in T_{p}M}$ .

Given a metric tensorg on ann-dimensional real manifold, thequadratic formq(x) =g(x,x) associated with the metric tensor applied to each vector of anyorthogonal basis producesn real values. BySylvester'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. Thesignature(p,q,r) of the metric tensor gives these numbers, shown in the same order. A non-degenerate metric tensor hasr = 0 and the signature may be denoted(p,q), wherep +q =n.

Apseudo-Riemannian manifold(M,g) is adifferentiable manifoldM that is equipped with an everywhere non-degenerate, smooth, symmetricmetric tensorg.

Such a metric is called apseudo-Riemannian metric. Applied to a vector field, the resulting scalar field value at any point of the manifold can be positive, negative or zero.

The signature of a pseudo-Riemannian metric is(p, q), where bothp andq are non-negative. The non-degeneracy condition together with continuity implies thatp andq remain unchanged throughout the manifold (assuming it is connected).

ALorentzian manifold is an important special case of a pseudo-Riemannian manifold in which thesignature of the metric is(1, n−1) (equivalently,(n−1, 1); seeSign convention). Such metrics are calledLorentzian metrics. They are named after the Dutch physicistHendrik Lorentz.

After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important in applications ofgeneral relativity.

A principal premise of general relativity is thatspacetime 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 intotimelike,null orspacelike. With a signature of(p, 1) or(1, q), the manifold is also locally (and possibly globally) time-orientable (seeCausal structure).

Just asEuclidean space${\displaystyle {\mathbb{R}}^n}$  can be thought of as the local model of aRiemannian manifold,Minkowski space${\displaystyle {\mathbb{R}}^{n-1,1}}$  with the flatMinkowski metric is the local model of a Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature (p, q) ispseudo-Euclidean space${\displaystyle {\mathbb{R}}^{p,q}}$ , for which there exist coordinatesx_i such that

${\displaystyle g=d{x}_1^2+\cdots +d{x}_p^2-d{x}_{p+1}^2-\cdots -d{x}_{p+q}^2}$ 

Some theorems of Riemannian geometry can be generalized to the pseudo-Riemannian case. In particular, thefundamental theorem of Riemannian geometry is true of all pseudo-Riemannian manifolds. This allows one to speak of theLevi-Civita connection on a pseudo-Riemannian manifold along with the associatedcurvature tensor. On the other hand, there are many theorems in Riemannian geometry that do not hold in the generalized case. For example, it isnot true that every smooth manifold admits a pseudo-Riemannian metric of a given signature; there are certaintopological obstructions. Furthermore, asubmanifold does not always inherit the structure of a pseudo-Riemannian manifold; for example, the metric tensor becomes zero on anylight-likecurve. TheClifton–Pohl torus provides an example of a pseudo-Riemannian manifold that is compact but not complete, a combination of properties that theHopf–Rinow theorem disallows for Riemannian manifolds.^[3]

• Causality conditions
• Globally hyperbolic manifold
• Hyperbolic partial differential equation
• Orientable manifold
• Spacetime

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• 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
• O'Neill, Barrett (1983),Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics, vol. 103, Academic Press,ISBN 9780080570570
• Vrănceanu, G.; Roşca, R. (1976),Introduction to Relativity and Pseudo-Riemannian Geometry, Bucarest: Editura Academiei Republicii Socialiste România

•   Media related toLorentzian manifolds at Wikimedia Commons