Pseudo Riemannian Manifold

Information about Pseudo Riemannian Manifold

In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. 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. Instead a weaker condition of nondegeneracy is imposed.

Arguably, the most important type of pseudo-Riemannian manifold is a Lorentzian manifold. Lorentzian manifolds occur in the general theory of relativity as models of curved 4-dimensional spacetime. Just as Riemannian manifolds may be thought of as being locally modeled on Euclidean space, Lorentzian manifolds are locally modeled on Minkowski space.

Formal definition

A pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric $\text{[math]}$ tensor which is nondegenerate at each point on the manifold. This tensor is called a pseudo-Riemannian metric or, simply, a (pseudo-)metric tensor.

Every nondegenerate, symmetric, bilinear form on a vector space can be assigned a signature $\text{[math]}$ . Here $\text{[math]}$ and $\text{[math]}$ denote the number of positive and negative eigenvalues of the form. The signature of a pseudo-Riemannian manifold is just the signature of the metric on any given tangent space (one should insist that the signature is the same on every connected component). Note that $\text{[math]}$ is the dimension of the manifold. A Riemannian metric has signature $\text{[math]}$ .

Lorentzian manifolds

Pseudo-Riemannian metrics of signature $\text{[math]}$ (or sometimes $\text{[math]}$ , see sign convention) are called Lorentzian metrics. A manifold equipped with a Lorentzian metric is naturally called a Lorentzian manifold. 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. A principal assumption of general relativity is that spacetime can be modeled as a Lorentzian manifold of signature $\text{[math]}$ .

Properties of pseudo-Riemannian manifolds

Just as Euclidean space Rⁿ can be thought of as the model Riemannian manifold, Minkowski space R^p,1 with the flat Minkowski metric is the model Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature $\text{[math]}$ is R^p,q with the metric

$\text{[math]}$

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.

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It arises naturally from the study of the theory of differential equations. Differential geometry is the study of geometry using differential calculus (cf.
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In Riemannian geometry, a Riemannian manifold (M,g) (with Riemannian metric g) is a real differentiable manifold M in which each tangent space is equipped with an inner product g
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Bernhard Riemann

Bernhard Riemann, 1863
Born September 17, 1826
Breselenz, Germany
Died July 20 1866 (aged 41)
Selasca, Italy
Residence Germany
Citizenship German
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In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. In other terms, given a smooth manifold, we make a choice of positive-definite quadratic form on the manifold's tangent spaces which varies
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In mathematics, a definite bilinear form is a bilinear form B such that

B(x, x)

has a fixed sign (positive or negative) when x is not 0.
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In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that the map from V to V* (the dual space of V) given by v f(-,v) is not a bijection.
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General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16.^[1] It unifies special relativity, Newton's law of universal gravitation, and the insight that gravitational
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spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of the fourth dimension.
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Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
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In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated.
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Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus.
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The term tensor has slightly different meanings in mathematics and physics. In the mathematical fields of multilinear algebra and differential geometry, a tensor is a multilinear function.
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In mathematics, a bilinear form on a vector space V is a bilinear mapping V Ã— V → F, where F is the field of scalars.
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In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are
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This article or section may be confusing or unclear for some readers.
Please [improve the article] or discuss this issue on the talk page. This article has been tagged since December 2006.
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eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue.
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In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the disjoint union of two or more nonempty open spaces.
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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. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set
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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.
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In Riemannian geometry, the Levi-Civita connection is the torsion-free Riemannian connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.
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The term curvature tensor is ambiguous in its generality. It could refer to:

the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds;
the curvature of an affine connection or covariant derivative (on tensors);

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Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure.
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