ARiemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

 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

Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

 f defined on U such that (U, e^2fg) is flat

Flat manifold

In mathematics, a Riemannian manifold is said to be flat if its curvature is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°....

 (i.e. the curvature

Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...

 of e^2fg 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.

Examples

• Every manifold with constant
  Constant curvature
  In mathematics, constant curvature in differential geometry is a concept most commonly applied to surfaces. For those the scalar curvature is a single number determining the local geometry, and its constancy has the obvious meaning that it is the same at all points...
  
   sectional curvature
  Sectional curvature
  In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...
  
   is conformally flat.
• Every 2-dimensional pseudo-Riemannian manifold is conformally flat.
• A 3-dimensional pseudo-Riemannian manifold is conformally flat if and only the Cotton tensor
  Cotton tensor
  In differential geometry, the Cotton tensor on a -Riemannian manifold of dimension n is a third-order tensor concomitant of the metric, like the Weyl tensor. The concept is named after Émile Cotton...
  
   vanishes.
• An n-dimensional pseudo-Riemannian manifold for n ≥ 4 is conformally flat if and only if the Weyl tensor
  Weyl tensor
  In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic...
  
   vanishes.
• Every compact
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
  , simply connected, conformally flat Riemannian manifold is conformally equivalent to the round sphere.