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Conformally flat manifold
From Wikipedia, the free encyclopedia
A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation.
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In practice, the metric of the manifold
has to be conformal to the flat metric
, i.e., the geodesics maintain in all points of
the angles by moving from one to the other, as well as keeping the null geodesics unchanged,[1] that means there exists a function
such that
, where
is known as the conformal factor and
is a point on the manifold.
More formally, let be a pseudo-Riemannian manifold. Then
is conformally flat if for each point
in
, there exists a neighborhood
of
and a smooth function
defined on
such that
is flat (i.e. the curvature of
vanishes on
). The function
need not be defined on all of
.
Some authors use the definition of locally conformally flat when referred to just some point on
and reserve the definition of conformally flat for the case in which the relation is valid for all
on
.