# 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.