Editing Conformal geometry (section)

====Representative metrics====
A representative [[Riemannian metric]] on the sphere is a metric which is proportional to the standard sphere metric.  This gives a realization of the sphere as a [[conformal geometry#Conformal manifolds|conformal manifold]].  The standard sphere metric is the restriction of the Euclidean metric on '''R'''<sup>''n''+1</sup>

:<math>g=dz^2+dx_1^2+dx_2^2+\cdots+dx_n^2</math>

to the sphere

:<math>z^2+x_1^2+x_2^2+\cdots+x_n^2=1.</math>

A conformal representative of ''g'' is a metric of the form ''λ''<sup>2</sup>''g'', where ''λ'' is a positive function on the sphere.  The conformal class of ''g'', denoted [''g''], is the collection of all such representatives:

:<math> [ g ] = \left\{ \lambda ^2 g \mid \lambda > 0 \right\} .</math>

An embedding of the Euclidean sphere into ''N''<sup>+</sup>, as in the previous section, determines a conformal scale on ''S''.  Conversely, any conformal scale on ''S'' is given by such an embedding.  Thus the line bundle {{nowrap|''N''<sup>+</sup> → ''S''}} is identified with the bundle of conformal scales on ''S'': to give a section of this bundle is tantamount to specifying a metric in the conformal class [''g''].