conformally

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



g


{\displaystyle g}

of the manifold



M


{\displaystyle M}

has to be conformal to the flat metric



η


{\displaystyle \eta }

, i.e., the geodesics maintain in all points of



M


{\displaystyle M}

the angles by moving from one to the other, as well as keeping the null geodesics unchanged, that means there exists a function



λ
(
x
)


{\displaystyle \lambda (x)}

such that



g
(
x
)
=

λ

2


(
x
)

η


{\displaystyle g(x)=\lambda ^{2}(x)\,\eta }

, where



λ
(
x
)


{\displaystyle \lambda (x)}

is known as the conformal factor and



x


{\displaystyle x}

is a point on the manifold.
More formally, let



(
M
,
g
)


{\displaystyle (M,g)}

be a pseudo-Riemannian manifold. Then



(
M
,
g
)


{\displaystyle (M,g)}

is conformally flat if for each point



x


{\displaystyle x}

in



M


{\displaystyle M}

, there exists a neighborhood



U


{\displaystyle U}

of



x


{\displaystyle x}

and a smooth function



f


{\displaystyle f}

defined on



U


{\displaystyle U}

such that



(
U
,

e

2
f


g
)


{\displaystyle (U,e^{2f}g)}

is flat (i.e. the curvature of




e

2
f


g


{\displaystyle e^{2f}g}

vanishes on



U


{\displaystyle U}

). The function



f


{\displaystyle f}

need not be defined on all of



M


{\displaystyle M}

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



x


{\displaystyle x}

on



M


{\displaystyle M}

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



x


{\displaystyle x}

on



M


{\displaystyle M}

.

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