# nLab anti de Sitter spacetime

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Gravity

gravity, supergravity

# Contents

## Definition

Up to isometry, the anti de Sitter spacetime of dimension $d + 1$ is the pseudo-Riemannian manifold whose underlying manifold is the submanifold of the Cartesian space $\mathbb{R}^{d+2}$ that solves the equation

$\sum_{i = 1}^{d+1} (x^i)^2 - (x^{d+2})^2 = 0$

and equipped with the metric induced from the ambient metric

$g = \sum_{i = 1}^{d+1} d x^i \otimes d x^i - d x^{i+1} \otimes d x^{i+t} \,,$

where $x^i\colon \mathbb{R}^{d+2} \to \mathbb{R}$ denote the canonical coordinates on a Cartesian space.

## Properties

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### Holography

Asymptotically ant-de Sitter spaces play a central role in the realization of the holographic principle by AdS/CFT correspondence.

## References

Revised on March 25, 2015 13:40:19 by Urs Schreiber (195.113.30.252)