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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation. 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 (U, e2fg) is flat (i.e. the curvature of e2fg vanishes on U). The function f need not be defined on all of M. Some authors use locally conformally flat to describe the above notion and reserve conformally flat for the case in which the function f is defined on all of M.