In

`p`, `q`) is $\backslash mathbb^$ with the metric
:$g\; =\; dx\_1^2\; +\; \backslash cdots\; +\; dx\_p^2\; -\; dx\_^2\; -\; \backslash cdots\; -\; dx\_^2$
Some basic theorems of Riemannian geometry can be generalized to the pseudo-Riemannian case. In particular, the

differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...

, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

with a metric tensor
In the mathematical
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that is everywhere nondegenerate
In mathematics, a degenerate case is a limiting case (mathematics), limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of ...

. This is a generalization of a Riemannian manifold
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integr ...

in which the requirement of positive-definiteness is relaxed.
Every tangent space
In mathematics
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of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space.
A special case used in general relativity
General relativity, also known as the general theory of relativity, is the geometric
Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...

is a four-dimensional Lorentzian manifold for modeling spacetime
In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...

, where tangent vectors can be classified as timelike, null, and spacelike.
Introduction

Manifolds

Indifferential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...

, a differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

is a space which is locally similar to a Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the coordinate
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

s 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 patch
In mathematics
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es: subsets of the manifold which can be mapped into ''n''-dimensional Euclidean space.
See ''Manifold
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

'', ''Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

'', ''Coordinate patch
In mathematics
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'' for more details.
Tangent spaces and metric tensors

Associated with each point $p$ in an $n$-dimensional differentiable manifold $M$ is atangent space
In mathematics
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(denoted $T\_pM$). This is an $n$-dimensional vector space
In mathematics
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whose elements can be thought of as equivalence class
In mathematics
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es of curves passing through the point $p$.
A metric tensor
In the mathematical
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is a non-degenerate
In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'') given by is not an isomorphism. An equivalent definition when ...

, smooth, symmetric, bilinear map
In mathematics
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that assigns a real number
In mathematics
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to pairs of tangent vectors at each tangent space of the manifold. Denoting the metric tensor by $g$ we can express this as
:$g\; :\; T\_pM\; \backslash times\; T\_pM\; \backslash to\; \backslash mathbb.$
The map is symmetric and bilinear so if $X,Y,Z\; \backslash in\; T\_pM$ are tangent vectors at a point $p$ to the manifold $M$ then we have
* $\backslash ,g(X,Y)\; =\; g(Y,X)$
* $\backslash ,g(aX\; +\; Y,\; Z)\; =\; a\; g(X,Z)\; +\; g(Y,Z)$
for any real number $a\backslash in\backslash mathbb$.
That $g$ is non-degenerate
In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'') given by is not an isomorphism. An equivalent definition when ...

means there are no non-zero $X\; \backslash in\; T\_pM$ such that $\backslash ,g(X,Y)\; =\; 0$ for all $Y\; \backslash in\; T\_pM$.
Metric signatures

Given a metric tensor ''g'' on an ''n''-dimensional real manifold, thequadratic form
In mathematics
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associated with the metric tensor applied to each vector of any orthogonal basis In mathematics
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produces ''n'' real values. By Sylvester's law of inertia
Sylvester's law of inertia is a theorem
In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a ...

, 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
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a s ...

of the metric tensor gives these numbers, shown in the same order. A non-degenerate metric tensor has and the signature may be denoted (''p'', ''q''), where .
Definition

A pseudo-Riemannian manifold $(M,g)$ is adifferentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

$M$ equipped with an everywhere non-degenerate, smooth, symmetric metric tensor
In the mathematical
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$g$.
Such a metric is called a pseudo-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 , where both ''p'' and ''q'' are non-negative. The non-degeneracy condition together with continuity implies that ''p'' and ''q'' remain unchanged throughout the manifold (assuming it is connected).
Lorentzian manifold

A Lorentzian manifold is an important special case of a pseudo-Riemannian manifold in which the signature of the metric is (equivalently, ; see ''Sign convention In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succes ...

''). Such metrics are called Lorentzian metrics. They are named after the Dutch physicist Hendrik Lorentz
Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist
A physicist is a scientist
A scientist is a person who conducts scientific research
The scientific method is an Empirical evidence, empirical met ...

.
Applications in physics

After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important in applications ofgeneral relativity
General relativity, also known as the general theory of relativity, is the geometric
Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...

.
A principal premise of general relativity is that spacetime
In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...

can be modeled as a 4-dimensional Lorentzian manifold of signature or, equivalently, . 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 or , the manifold is also locally (and possibly globally) time-orientable (see ''Causal structure
In mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in ...

'').
Properties of pseudo-Riemannian manifolds

Just asEuclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

$\backslash mathbb^n$ can be thought of as the model Riemannian manifold
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integr ...

, Minkowski space
In mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...

$\backslash mathbb^$ with the flat Minkowski metric
In mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...

is the model Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature (fundamental theorem of Riemannian geometry
Fundamental may refer to:
* Foundation of reality
Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only imaginary
Imaginary may refer to:
* Imaginary (sociology), a concept in socio ...

is true of pseudo-Riemannian manifolds as well. This allows one to speak of the Levi-Civita connection
In Riemannian manifold, Riemannian or pseudo-Riemannian manifold, pseudo Riemannian geometry (in particular the Lorentzian manifold, Lorentzian geometry of General Relativity, general relativity), the Levi-Civita connection is the unique affine co ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

obstructions. Furthermore, a submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...

does not always inherit the structure of a pseudo-Riemannian manifold; for example, the metric tensor becomes zero on any light-like
In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...

curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

. The Clifton–Pohl torus In geometry, the Clifton–Pohl torus is an example of a compact space, compact Lorentzian manifold that is not geodesically complete. While every compact Riemannian manifold is also geodesically complete (by the Hopf–Rinow theorem), this space sh ...

provides an example of a pseudo-Riemannian manifold that is compact but not complete, a combination of properties that the Hopf–Rinow theorem
Hopf–Rinow theorem is a set of statements about the Geodesic manifold, geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931.
Statement
Let (M, g) be a Connected space, ...

disallows for Riemannian manifolds., p. 193.
See also

*Causality conditions
In the study of Lorentzian manifold
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study p ...

*Globally hyperbolic manifold
In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold ...

*Hyperbolic partial differential equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

*Orientable manifold
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

*Spacetime
In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...

Notes

References

* * * * *.External links

* {{Riemannian geometry Bernhard Riemann Differential geometry * Riemannian geometry Riemannian manifolds Smooth manifolds