Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M It is one of many things named after Bernhard Riemann. The key difference between the two is that on a pseudo-Riemannian manifold the metric tensor need not be positive-definite. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Mathematics, a definite bilinear form is a Bilinear form B such that B ( x, x) has a fixed Instead a weaker condition of nondegeneracy is imposed. In Mathematics, specifically Linear algebra, a degenerate Bilinear form f(xy on a Vector space V is one such that

1 Introduction
2 Definition
3 Lorentzian manifold
- 3.1 Applications in physics
4 Properties of pseudo-Riemannian manifolds
5 See also

Introduction

Manifolds

Main articles: Manifold, differentiable manifolds

In differential geometry a differentiable manifold is a space which is locally similar to a Euclidean space. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In an $n$ -dimensional Euclidean space any point can be specified by $n$ real numbers. These are called the coordinates of the point. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each 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. For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how

See Manifold, differentiable manifold, coordinate patch for more details. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how

Tangent spaces and metric tensors

Main articles: Tangent space, metric tensor

Associated with each point $p$ in an $n$ -dimensional differentiable manifold $M$ is a tangent space (denoted $\,T_pM$ ). In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since This is an $n$ -dimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point $p$ . In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X

A metric tensor is a non-degenerate, smooth, symmetric, bilinear map which assigns a real number to pairs of tangent vectors at each tangent space of the manifold. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Mathematics, specifically Linear algebra, a degenerate Bilinear form f(xy on a Vector space V is one such that In Mathematics, a bilinear map is a function of two arguments that is linear in each In Mathematics, the real numbers may be described informally in several different ways Denoting the metric tensor by $g$ we can express this as $g : T_pM \times T_pM \to \mathbb{R}$ .

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

$\,g(X,Y) = g(Y,X)$
$\,g(aX + Y, Z) = a g(X,Z) + g(Y,Z)$

for some real number $a$ .

That $g$ is non-degenerate means there are no non-zero $X \in T_pM$ such that $\,g(X,Y) = 0$ for all $Y \in T_pM$ . In Mathematics, specifically Linear algebra, a degenerate Bilinear form f(xy on a Vector space V is one such that

Metric signatures

Main article: Metric signature

For an $n$ -dimensional manifold the metric tensor (in a fixed coordinate system) has $n$ eigenvalues. The signature of a Metric tensor (or more generally a nondegenerate Symmetric bilinear form, thought of as Quadratic form) is the number of positive In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes If the metric is non-degenerate then none of these eigenvalues are zero. The signature of the metric denotes the number of positive and negative eigenvalues, this quantity is independent of the chosen coordinate system by Sylvester's rigidity theorem and locally non-decreasing. The signature of a Metric tensor (or more generally a nondegenerate Symmetric bilinear form, thought of as Quadratic form) is the number of positive If the metric has $p$ positive eigenvalues and $q$ negative eigenvalues then the metric signature is $(p, q)$ . For a non-degenerate metric $p + q = n$ .

Definition

A pseudo-Riemannian manifold $\,(M,g)$ is a differentiable manifold $\,M$ equipped with a non-degenerate, smooth, symmetric metric tensor $\,g$ which, unlike a Riemannian metric, need not be positive-definite, but must be non-degenerate. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, a definite bilinear form is a Bilinear form B such that B ( x, x) has a fixed Such a metric is called a pseudo-Riemannian metric and its values can be positive, negative or zero.

The signature of a pseudo-Riemannian metric is $(p, q)$ where both $p$ and $q$ are non-zero.

Lorentzian manifold

A Lorentzian manifold is an important special case of a pseudo-Riemannian manifold in which the signature of the metric is $(1, n - 1)$ (or sometimes $(n - 1,1)$ , see sign convention). In Physics, a sign convention is a choice of the signs (plus or minus of a set of quantities in a case where the choice of sign is arbitrary Such metrics are called Lorentzian metrics. They are named after the physicist Hendrik Lorentz. Hendrik Antoon Lorentz ( July 18, 1853 &ndash February 4, 1928) was a Dutch Physicist who shared the 1902 Nobel

Applications in physics

After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important because of their physical applications to the theory of general relativity. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

A principal assumption of general relativity is that spacetime can be modeled as a 4-dimensional Lorentzian manifold of signature $(3,1)$ (or equivalently (1,3)). General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Unlike Riemannian manifolds with positive-definite metrics, a signature of $(p,1)$ or $(1, q)$ allows tangent vectors to be classified into timelike, null or spacelike (see Causal structure). The causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold

Properties of pseudo-Riemannian manifolds

Just as Euclidean space $\mathbb{R}^n$ can be thought of as the model Riemannian manifold, Minkowski space $\mathbb{R}^{n-1,1}$ with the flat Minkowski metric is the model Lorentzian manifold. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity Likewise, the model space for a pseudo-Riemannian manifold of signature $(p, q)$ is $\mathbb{R}^{p,q}$ with the metric: $g = dx_1^2 + \cdots + dx_p^2 - dx_{p+1}^2 - \cdots - dx_n^2$

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. In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or Pseudo-Riemannian manifold) there is a This allows one to speak of the Levi-Civita connection on a pseudo-Riemannian manifold along with the associated curvature tensor. In Riemannian geometry, the Levi-Civita connection is the torsion -free Riemannian connection, i 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. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Furthermore, a submanifold of a pseudo-Riemannian manifold need not be a pseudo-Riemannian manifold. 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

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