Conformally flat manifold

A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation.

In practice, the metric

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

is known as the conformal factor and

is a point on the manifold.

More formally, let

be a pseudo-Riemannian manifold.

is conformally flat if for each point

, there exists a neighborhood

and a smooth function

is flat (i.e. the curvature of

Some authors use the definition of locally conformally flat when referred to just some point

and reserve the definition of conformally flat for the case in which the relation is valid for all

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