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
![]() | (*) |
where
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For , every
is a conformal Euclidean space. In order that a space with
be a conformal Euclidean space, it is necessary and sufficient that there exist a tensor
satisfying the conditions (*) and
. 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) |
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 be an
-dimensional Riemannian space with Riemannian metric
, Levi-Civita derivation (cf. Levi-Civita connection)
, curvature tensor
, Ricci transformation (cf. Ricci tensor)
, and scalar curvature
. Then the conformal curvature tensor
(Weyl's curvature tensor) is defined by
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where
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and
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Then locally admits a conformal mapping onto some open set of
if and only if
1) for
; or
2) and
for
.
(See [a1] for example; for the "Codazzi equationCodazzi equation" for
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) |
Conformal Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conformal_Euclidean_space&oldid=14002