Riemannian geometry | Manifolds | Conformal geometry
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 of the manifold 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, that means exists a function such that , where is known as the conformal factor and is a point on the manifold. More formally, let be a pseudo-Riemannian manifold. Then is conformally flat if for each point in , there exists a neighborhood of and a smooth function defined on such that is flat (i.e. the curvature of vanishes on ). The function need not be defined on all of . Some authors use the definition of locally conformally flat when referred to just some point on and reserve the definition of conformally flat for the case in which the relation is valid for all on . (Wikipedia).
