Pseudo-Riemannian manifold

pseudo-Riemannianpseudo-Riemannian metricpseudoLorentz metricpseudo-Riemannian geometryLorentzian manifoldpseudo-Lorentzian manifoldsLorentzian spacetimesemi-Riemannian geometry
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate.wikipedia
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Differential geometry

differentialdifferential geometerdifferential geometry and topology
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space.
Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite.

Spacetime

space-timespace-time continuumspace and time
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.
Minkowski's geometric interpretation of relativity was to prove vital to Einstein's development of his 1915 general theory of relativity, wherein he showed how mass and energy curve flat spacetime into a pseudo-Riemannian manifold.

Metric tensor

metricmetricsround metric
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. 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.
have constant signature independent of m, then the signature of g is this signature, and g is called a pseudo-Riemannian metric.

General relativity

general theory of relativitygeneral relativity theoryrelativity
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.
The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric.

Differentiable manifold

smooth manifoldsmoothdifferential manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. See Manifold, Differentiable manifold, Coordinate patch for more details.
A pseudo-Riemannian manifold is a variant of Riemannian manifold where the metric tensor is allowed to have an indefinite signature (as opposed to a positive-definite one).

Pseudo-Euclidean space

pseudo-Euclideanpseudo-Euclidean planepseudo-Euclidean vector space
Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space.
The correspondence between contravariant and covariant tensors makes a tensor calculus on pseudo-Riemannian manifolds analogous to one on Riemannian manifolds.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. 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.

Metric signature

signatureSignature changeindefinite signature
The signature (p, q, r) of the metric tensor gives these numbers, shown in the same order.
The Lorentzian metric is a metric signature (v, p) with two eigenvalues.

Degenerate bilinear form

non-degeneratenondegeneratedegenerate
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. 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.
For example, a manifold with an inner product structure on its tangent spaces is a Riemannian manifold, while relaxing this to a symmetric nondegenerate form yields a pseudo-Riemannian manifold.

Fundamental theorem of Riemannian geometry

fundamental theorems of Riemannian geometry
In particular, the fundamental theorem of Riemannian geometry is true of pseudo-Riemannian manifolds as well.
In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric.

Levi-Civita connection

Christoffel symbolconnectionsLevi-Civita
This allows one to speak of the Levi-Civita connection on a pseudo-Riemannian manifold along with the associated curvature tensor.
More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.

Riemann curvature tensor

Riemann tensorcurvature tensorcurvature
This allows one to speak of the Levi-Civita connection on a pseudo-Riemannian manifold along with the associated curvature tensor.
The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

Euclidean space

EuclideanspaceEuclidean vector space
In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. 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.
To take the gravity into account, general relativity uses a pseudo-Riemannian manifold that has Minkowski spaces as tangent spaces.

Minkowski space

Minkowski spacetimeMinkowski metricflat spacetime
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.
. It is perhaps the simplest example of a pseudo-Riemannian manifold.

Orthogonal basis

orthogonalorthogonal basesorthogonal basis set
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.
Any orthogonal basis can be used to define a system of orthogonal coordinates V. Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates in Euclidean spaces, as well as in Riemannian and pseudo-Riemannian manifolds.

Definite quadratic form

positive definitepositive-definitepositive semi-definite
This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.

Tangent space

tangent planetangenttangent vector
Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space.

Causal structure

causal relationcausal curvenull
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.

Coordinate system

coordinatescoordinateaxis
These are called the coordinates of the point.

Atlas (topology)

atlascharttransition map
This is achieved by defining coordinate patches: subsets of the manifold which can be mapped into n-dimensional Euclidean space. See Manifold, Differentiable manifold, Coordinate patch for more details.

Manifold

manifoldsboundarymanifold with boundary
See Manifold, Differentiable manifold, Coordinate patch for more details.

Vector space

vectorvector spacesvectors
This is an n-dimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point p.

Equivalence class

quotient setequivalence classesquotient
This is an n-dimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point p.

Bilinear map

bilinearbilinearitybilinear operator
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.

Real number

realrealsreal-valued
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.