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Ambient construction, the Glossary

Index Ambient construction

In conformal geometry, the ambient construction refers to a construction of Charles Fefferman and Robin Graham for which a conformal manifold of dimension n is realized (ambiently) as the boundary of a certain Poincaré manifold, or alternatively as the celestial sphere of a certain pseudo-Riemannian manifold.[1]

Table of Contents

  1. 26 relations: AdS/CFT correspondence, Asymptote, Bach tensor, C. Robin Graham, Celestial sphere, Charles Fefferman, Conformal connection, Conformal geometry, Differential operator, Dirichlet boundary condition, Fiber bundle, GJMS operator, Holographic principle, Hyperbolic manifold, Lie derivative, Line bundle, Minkowski space, Null vector, Obstruction theory, Order of approximation, Pseudo-Riemannian manifold, Pullback (differential geometry), Ricci-flat manifold, Schouten tensor, Tractor bundle, Weyl tensor.

  2. Conformal geometry
  3. Tensors in general relativity

AdS/CFT correspondence

In theoretical physics, the anti-de Sitter/conformal field theory correspondence (frequently abbreviated as AdS/CFT) is a conjectured relationship between two kinds of physical theories.

See Ambient construction and AdS/CFT correspondence

Asymptote

In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.

See Ambient construction and Asymptote

Bach tensor

In differential geometry and general relativity, the Bach tensor is a trace-free tensor of rank 2 which is conformally invariant in dimension. Ambient construction and Bach tensor are tensors in general relativity.

See Ambient construction and Bach tensor

C. Robin Graham

Charles Robin Graham is professor emeritus of mathematics at the University of Washington, known for a number of contributions to the field of conformal geometry and CR geometry; his collaboration with Charles Fefferman on the ambient construction has been particularly widely cited.

See Ambient construction and C. Robin Graham

Celestial sphere

In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth.

See Ambient construction and Celestial sphere

Charles Fefferman

Charles Louis Fefferman (born April 18, 1949) is an American mathematician at Princeton University, where he is currently the Herbert E. Jones, Jr.

See Ambient construction and Charles Fefferman

Conformal connection

In conformal differential geometry, a conformal connection is a Cartan connection on an n-dimensional manifold M arising as a deformation of the Klein geometry given by the celestial ''n''-sphere, viewed as the homogeneous space where P is the stabilizer of a fixed null line through the origin in Rn+2, in the orthochronous Lorentz group O+(n+1,1) in n+2 dimensions. Ambient construction and conformal connection are conformal geometry.

See Ambient construction and Conformal connection

Conformal geometry

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

See Ambient construction and Conformal geometry

Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

See Ambient construction and Differential operator

Dirichlet boundary condition

In mathematics, the Dirichlet boundary condition is imposed on an ordinary or partial differential equation, such that the values that the solution takes along the boundary of the domain are fixed.

See Ambient construction and Dirichlet boundary condition

Fiber bundle

In mathematics, and particularly topology, a fiber bundle (''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure.

See Ambient construction and Fiber bundle

GJMS operator

In the mathematical field of differential geometry, the GJMS operators are a family of differential operators, that are defined on a Riemannian manifold. Ambient construction and GJMS operator are conformal geometry.

See Ambient construction and GJMS operator

Holographic principle

The holographic principle is a property of string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region — such as a light-like boundary like a gravitational horizon.

See Ambient construction and Holographic principle

Hyperbolic manifold

In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension.

See Ambient construction and Hyperbolic manifold

Lie derivative

In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field.

See Ambient construction and Lie derivative

Line bundle

In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space.

See Ambient construction and Line bundle

Minkowski space

In physics, Minkowski space (or Minkowski spacetime) is the main mathematical description of spacetime in the absence of gravitation.

See Ambient construction and Minkowski space

Null vector

In mathematics, given a vector space X with an associated quadratic form q, written, a null vector or isotropic vector is a non-zero element x of X for which.

See Ambient construction and Null vector

Obstruction theory

In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.

See Ambient construction and Obstruction theory

Order of approximation

In science, engineering, and other quantitative disciplines, order of approximation refers to formal or informal expressions for how accurate an approximation is.

See Ambient construction and Order of approximation

Pseudo-Riemannian manifold

In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate.

See Ambient construction and Pseudo-Riemannian manifold

Pullback (differential geometry)

Let \phi:M\to N be a smooth map between smooth manifolds M and N. Then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is known as the pullback (by \phi), and is frequently denoted by \phi^*.

See Ambient construction and Pullback (differential geometry)

Ricci-flat manifold

In the mathematical field of differential geometry, Ricci-flatness is a condition on the curvature of a Riemannian manifold.

See Ambient construction and Ricci-flat manifold

Schouten tensor

In Riemannian geometry the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten defined for by: where Ric is the Ricci tensor (defined by contracting the first and third indices of the Riemann tensor), R is the scalar curvature, g is the Riemannian metric, J. Ambient construction and Schouten tensor are tensors in general relativity.

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Tractor bundle

In conformal geometry, the tractor bundle is a particular vector bundle constructed on a conformal manifold whose fibres form an effective representation of the conformal group (see associated bundle). Ambient construction and tractor bundle are conformal geometry.

See Ambient construction and Tractor bundle

Weyl tensor

In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Ambient construction and Weyl tensor are tensors in general relativity.

See Ambient construction and Weyl tensor

See also

Conformal geometry

Tensors in general relativity

References

[1] https://en.wikipedia.org/wiki/Ambient_construction

Also known as Ambient metric, Obstruction tensor.