# Conformal Euclidean space

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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 ).

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