# Pseudo-Riemannian manifold

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In differential geometry, a pseudo-Riemannian manifold, [1] [2] also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.

## Contents

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

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.

## Introduction

### Manifolds

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.

See Manifold , Differentiable manifold , Coordinate patch for more details.

### Tangent spaces and metric tensors

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

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

### Metric signatures

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 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. The signature (p, q, r) of the metric tensor gives these numbers, shown in the same order. A non-degenerate metric tensor has r = 0 and the signature may be denoted (p, q), where p + q = n.

## Definition

A pseudo-Riemannian manifold${\displaystyle (M,g)}$ is a differentiable manifold ${\displaystyle M}$ equipped with an everywhere non-degenerate, smooth, symmetric metric tensor ${\displaystyle g}$.

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 (p, q), 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 (1, n−1) (equivalently, (n−1, 1); see Sign convention ). Such metrics are called Lorentzian metrics. They are named after the Dutch physicist Hendrik Lorentz.

### Applications in physics

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

A principal premise 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, an indefinite signature allows tangent vectors to be classified into timelike, null or spacelike. With a signature of (p, 1) or (1, q), the manifold is also locally (and possibly globally) time-orientable (see Causal structure ).

## Properties of pseudo-Riemannian manifolds

Just as Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ can be thought of as the model Riemannian manifold, Minkowski space ${\displaystyle \mathbb {R} ^{n-1,1}}$ with the flat Minkowski metric is the model Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature (p, q) is ${\displaystyle \mathbb {R} ^{p,q}}$ with the metric

${\displaystyle g=dx_{1}^{2}+\cdots +dx_{p}^{2}-dx_{p+1}^{2}-\cdots -dx_{p+q}^{2}}$

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. [3]

## Related Research Articles

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In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold. It is a local invariant of Riemannian metrics which measure the failure of second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.

In differential geometry, a Riemannian manifold or Riemannian space(M, g) is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p. A common convention is to take g to be smooth, which means that for any smooth coordinate chart (U, x) on M, the n2 functions

In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.

In mathematical physics, Minkowski space is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity.

In Riemannian or pseudo Riemannian geometry, the Levi-Civita connection is the unique connection on the tangent bundle of a manifold that preserves the (pseudo-)Riemannian metric and is torsion-free.

In mathematics, an isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds with dimension greater than 2. The sectional curvature Kp) depends on a two-dimensional linear subspace σp of the tangent space at a point p of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σp as a tangent plane at p, obtained from geodesics which start at p in the directions of σp. The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

In mathematics, the signature(v, p, r) of a metric tensor g is the number of positive, negative and zero eigenvalues of the real symmetric matrix gab of the metric tensor with respect to a basis. In relativistic physics, the v represents the time or virtual dimension, and the p for the space and physical dimension. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis. The signature thus classifies the metric up to a choice of basis. The signature is often denoted by a pair of integers (v, p) implying r= 0, or as an explicit list of signs of eigenvalues such as (+, −, −, −) or (−, +, +, +) for the signatures (1, 3, 0) and (3, 1, 0), respectively.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations, although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to the four-dimensional Lorentzian manifolds usually studied in general relativity. Einstein manifolds in four Euclidean dimensions are studied as gravitational instantons.

In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc length, is a function of the metric tensor and is denoted by ds

In mathematics, a metric or distance function is a function that gives a distance between each pair of point elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable.

In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami.

In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choice of basis (e1, …, en), be applied to a vector x = x1e1 + ⋯ + xnen, giving

In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.

In gravitation theory, a world manifold endowed with some Lorentzian pseudo-Riemannian metric and an associated space-time structure is a space-time. Gravitation theory is formulated as classical field theory on natural bundles over a world manifold.

## References

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