# Conformal Euclidean space

A Riemannian space admitting a conformal mapping onto a Euclidean space. The curvature tensor of a conformal Euclidean space has the form

$$\tag{* } R _ {ijk.} ^ {l} = \ 2T _ {..k[i } ^ {lm} p _ {j]m } ,$$

where

$$T _ {..ij} ^ {km} = \ \delta _ {i} ^ {k} \delta _ {j} ^ {m} + \delta _ {j} ^ {k} \delta _ {i} ^ {m} - g ^ {km} g _ {ij} ,$$

$$p _ {ij} = \nabla _ {i} p _ {j} - { \frac{1}{2} } T _ {..ij } ^ {km} p _ {k} p _ {m} .$$

For $n = 2$, every $V _ {n}$ is a conformal Euclidean space. In order that a space with $n > 3$ be a conformal Euclidean space, it is necessary and sufficient that there exist a tensor $p _ {ij}$ satisfying the conditions (*) and $\nabla _ {[k } p _ {i]j } = 0$. Sometimes a conformal Euclidean space is called a Weyl space admitting a conformal mapping onto a Euclidean space (see [2]).

#### References

 [1] J.A. Schouten, D.J. Struik, "Einführung in die neueren Methoden der Differentialgeometrie" , 2 , Noordhoff (1935) [2] A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)

The notion defined in the article above is also called a conformally Euclidean space. An alternative description of this notion is as follows. Let $M$ be an $n$- dimensional Riemannian space with Riemannian metric $g$, Levi-Civita derivation (cf. Levi-Civita connection) $D$, curvature tensor $R$, Ricci transformation (cf. Ricci tensor) $\mathop{\rm Ric}$, and scalar curvature $K$. Then the conformal curvature tensor $C$( Weyl's curvature tensor) is defined by

$$C ( X, Y) Z = \ R ( X, Y) Z - ( X \wedge Y) ( L ( Z)) - L (( X \wedge Y) ( Z)) ,$$

where

$$L ( W) = \ { \frac{1}{n - 2 } } \mathop{\rm Ric} ( W) - \frac{K}{2 ( n - 1) ( n - 2) } W$$

and

$$( X \wedge Y) ( W) = \ g ( Y, W) X - g ( X, W) Y.$$

Then $M$ locally admits a conformal mapping onto some open set of $E ^ {n}$ if and only if

1) $C = 0$ for $n > 3$; or

2) $C = 0$ and $( D _ {X} L) ( Y) = ( D _ {Y} L) ( X)$ for $n = 3$.

(See [a1] for example; for $n > 3$ the "Codazzi equationCodazzi equation" for $L$ is satisfied automatically.) The coordinate expressions for the equations given above can be found in the book of J.A. Schouten [a2].

#### References

 [a1] K. Yano, "The theory of Lie derivatives and its applications" , North-Holland (1957) [a2] J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) (Translated from German)
How to Cite This Entry:
Conformal Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conformal_Euclidean_space&oldid=46452
This article was adapted from an original article by G.V. Bushmanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article