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)

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 $ 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)