View source for 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c0247501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c0247502.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c0247503.png" /></td> </tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c0247504.png" />, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c0247505.png" /> is a conformal Euclidean space. In order that a space with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c0247506.png" /> be a conformal Euclidean space, it is necessary and sufficient that there exist a tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c0247507.png" /> satisfying the conditions (*) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c0247508.png" />. Sometimes a conformal Euclidean space is called a Weyl space admitting a conformal mapping onto a Euclidean space (see [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.A. Schouten,   D.J. Struik,   "Einführung in die neueren Methoden der Differentialgeometrie" , '''2''' , Noordhoff  (1935)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.P. Norden,   "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR></table>



====Comments====
The notion defined in the article above is also called a conformally Euclidean space. An alternative description of this notion is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c0247509.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475010.png" />-dimensional [[Riemannian space|Riemannian space]] with [[Riemannian metric|Riemannian metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475011.png" />, Levi-Civita derivation (cf. [[Levi-Civita connection|Levi-Civita connection]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475012.png" />, [[Curvature tensor|curvature tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475013.png" />, Ricci transformation (cf. [[Ricci tensor|Ricci tensor]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475014.png" />, and [[Scalar curvature|scalar curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475015.png" />. Then the conformal curvature tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475016.png" /> (Weyl's curvature tensor) is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475017.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475018.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475019.png" /></td> </tr></table>

Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475020.png" /> locally admits a conformal mapping onto some open set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475021.png" /> if and only if

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475022.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475023.png" />; or

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475025.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475026.png" />.

(See [[#References|[a1]]] for example; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475027.png" /> the  "Codazzi equationCodazzi equation"  for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024750/c02475028.png" /> is satisfied automatically.) The coordinate expressions for the equations given above can be found in the book of J.A. Schouten [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Yano,   "The theory of Lie derivatives and its applications" , North-Holland  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.A. Schouten,   "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer  (1954)  (Translated from German)</TD></TR></table>