Editing Conformal geometry (section)

==Conformal manifolds==
A '''conformal manifold''' is a [[Riemannian manifold]] (or [[pseudo-Riemannian manifold]]) equipped with an [[equivalence class]] of [[metric tensor]]s, in which two metrics ''g'' and ''h'' are equivalent if and only if

:<math>h = \lambda^2 g ,</math>

where ''λ'' is a real-valued [[smooth function]] defined on the manifold and is called the '''conformal factor'''. An equivalence class of such metrics is known as a '''conformal metric''' or '''conformal class'''.  Thus, a conformal metric may be regarded as a metric that is only defined "up to scale".  Often conformal metrics are treated by selecting a metric in the conformal class, and applying only "conformally invariant" constructions to the chosen metric.

A conformal metric is '''[[Conformally flat manifold|conformally flat]]''' if there is a metric representing it that is flat, in the usual sense that the [[Riemann curvature tensor]] vanishes.  It may only be possible to find a metric in the conformal class that is flat in an open neighborhood of each point.  When it is necessary to distinguish these cases, the latter is called ''locally conformally flat'', although often in the literature no distinction is maintained. The [[n-sphere|''n''-sphere]] is a locally conformally flat manifold that is not globally conformally flat in this sense, whereas a Euclidean space, a torus, or any conformal manifold that is covered by an open subset of Euclidean space is (globally) conformally flat in this sense.  A locally conformally flat manifold is locally conformal to a [[Möbius geometry]], meaning that there exists an angle preserving [[local diffeomorphism]] from the manifold into a Möbius geometry. In two dimensions, every conformal metric is locally conformally flat. In dimension {{nowrap|''n'' > 3}} a conformal metric is locally conformally flat if and only if its [[Weyl tensor]] vanishes; in dimension {{nowrap|1=''n'' = 3}}, if and only if the [[Cotton tensor]] vanishes.

Conformal geometry has a number of features which distinguish it from (pseudo-)Riemannian geometry.  The first is that although in (pseudo-)Riemannian geometry one has a well-defined metric at each point, in conformal geometry one only has a class of metrics.  Thus the length of a [[tangent vector]] cannot be defined, but the angle between two vectors still can.  Another feature is that there is no [[Levi-Civita connection]] because if ''g'' and ''λ''<sup>2</sup>''g'' are two representatives of the conformal structure, then the [[Christoffel symbol]]s of ''g'' and ''λ''<sup>2</sup>''g'' would not agree.  Those associated with ''λ''<sup>2</sup>''g'' would involve derivatives of the function λ whereas those associated with ''g'' would not.

Despite these differences, conformal geometry is still tractable.  The Levi-Civita connection and [[curvature form|curvature tensor]], although only being defined once a particular representative of the conformal structure has been singled out, do satisfy certain transformation laws involving the ''λ'' and its derivatives when a different representative is chosen.  In particular, (in dimension higher than 3) the [[Weyl tensor]] turns out not to depend on ''λ'', and so it is a '''conformal invariant'''.  Moreover, even though there is no Levi-Civita connection on a conformal manifold, one can instead work with a [[conformal connection]], which can be handled either as a type of [[Cartan connection]] modelled on the associated Möbius geometry, or as a [[Weyl connection]].  This allows one to define '''conformal curvature''' and other invariants of the conformal structure.