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 formula_1 of the manifold formula_2 has to be conformal to the flat metric formula_3, i.e., the geodesics maintain in all points of formula_2 the angles by moving from one to the other, as well as keeping the null geodesics unchanged, that means exists a function formula_5 such that formula_6, where formula_5 is known as the conformal factor and formula_8 is a point on the manifold. More formally, let formula_9 be  a pseudo-Riemannian manifold. Then formula_9 is conformally flat if for each point formula_8 in formula_2, there exists a neighborhood formula_13 of formula_8 and a smooth function formula_15 defined on formula_13 such that formula_17 is flat (i.e. the curvature of formula_18 vanishes on formula_13). The function formula_15 need not be defined on all of formula_2. Some authors use the definition of locally conformally flat when referred to just some point formula_8 on formula_2 and reserve the definition of "conformally flat" for the case in which the relation is valid for all formula_8 on formula_2.