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 of the manifold M has to be conformal to the flat metric \eta, i.e., the geodesics maintain in all points of M the angles by moving from one to the other, as well as keeping the null geodesics unchanged, that means there exists a function \lambda(x) such that, where \lambda(x) is known as the conformal factor and x is a point on the manifold. More formally, let (M,g) be a pseudo-Riemannian manifold. Then (M,g) is conformally flat if for each point x in M, there exists a neighborhood U of x and a smooth function f defined on U such that is flat (i.e. the curvature of e^{2f} g vanishes on U). The function f need not be defined on all of M. Some authors use the definition of locally conformally flat when referred to just some point x on M and reserve the definition of conformally flat for the case in which the relation is valid for all x on M.

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