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 tensor g {\displaystyle g} of the manifold M {\displaystyle M} has to be conformal to the flat metric tensor 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.

Source: Wikipedia — Conformally flat manifold (CC BY-SA 4.0)