# nLab conformal transformation

### Context

#### Riemannian geometry

Riemannian geometry

## Basic definitions

• Riemannian manifold

• moduli space of Riemannian metrics

• pseudo-Riemannian manifold

• geodesic

• Levi-Civita connection

• ## Theorems

• Poincaré conjecture-theorem
• ## Applications

• gravity

• # Contents

## Definition

A conformal transformation is a diffeomorphism that preserves a conformal structure (a conformal map). The group of conformal transformations is the conformal group.

## Properties

• A bijective conformal transformation of the Riemann sphere (to itself) is also called a Möbius transformation.

• For $p + q \geq 3$ then the group of conformal transformations of $\mathbb{R}^{p,q}$ is $\simeq O(p+1,q+1)$

## References

• Wolfgang Kühnel, Hans-Bert Rademacher, Liouville’s theorem in conformal geometry (pdf)

• Isadore Singer, Shlomo Sternberg, The infinite groups of Lie and Cartan. J. Anal. Math. 15, 1-114 (1965)

briefly reviewed in

• Shlomo Sternberg, Conformally flat geometry and supergeometry, section 3 of On charge conjugation, Comm. Math. Phys. Volume 109, Number 4 (1987), 649-679. (Euclid)

Last revised on July 10, 2018 at 12:33:02. See the history of this page for a list of all contributions to it.