Conformal geometry
It may only be possible to find a metric in the conformal class that is flat in an open neighborhood of each point.
When it is necessary to distinguish these cases, the latter is called locally conformally flat, although often in the literature no distinction is maintained.
Another feature is that there is no Levi-Civita connection because if g and λ2g are two representatives of the conformal structure, then the Christoffel symbols of g and λ2g would not agree.
In particular, (in dimension higher than 3) the Weyl tensor turns out not to depend on λ, and so it is a conformal invariant.
That is, the setting is a compactification of a familiar space; the geometry is concerned with the implications of preserving angles.
At an abstract level, the Euclidean and pseudo-Euclidean spaces can be handled in much the same way, except in the case of dimension two.
The compactified two-dimensional Minkowski plane exhibits extensive conformal symmetry.
equipped with the metric A 1-parameter group of conformal transformations gives rise to a vector field X with the property that the Lie derivative of g along X is proportional to g. Symbolically, In particular, using the above description of the Lie algebra cso(1, 1), this implies that for some real-valued functions a and b depending, respectively, on x and y. Conversely, given any such pair of real-valued functions, there exists a vector field X satisfying 1. and 2.
The comparative lack of rigidity of the two-dimensional case with that of higher dimensions owes to the analytical fact that the asymptotic developments of the infinitesimal automorphisms of the structure are relatively unconstrained.
In the case of higher dimensions, the asymptotic developments of infinitesimal symmetries are at most quadratic polynomials.
[3] The general theory of conformal geometry is similar, although with some differences, in the cases of Euclidean and pseudo-Euclidean signature.
[4] In either case, there are a number of ways of introducing the model space of conformally flat geometry.
Unless otherwise clear from the context, this article treats the case of Euclidean conformal geometry with the understanding that it also applies, mutatis mutandis, to the pseudo-Euclidean situation.
[5] From this perspective, the transformation properties of flat conformal space are those of inversive geometry.
In a related construction, the quadric S is thought of as the celestial sphere at infinity of the null cone in the Minkowski space Rn+1,1, which is equipped with the quadratic form q as above.
On the other hand, Riemannian isometries of a sphere are generated by inversions in geodesic hyperspheres (see the Cartan–Dieudonné theorem.)
For pseudo-Euclidean of metric signature (p, q), the model flat geometry is defined analogously as the homogeneous space O(p + 1, q + 1)/H, where H is again taken as the stabilizer of a null line.
The stabilizer of the null ray pointing up the last coordinate vector is given by the Borel subalgebra
