Parabolic geometry

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In mathematics, the term parabolic geometry is used in the following meanings:

• Euclidean geometry
• More generally, a Riemannian manifold having zero scalar curvature (the word elliptic would indicate positive scalar curvature and the word hyperbolic indicates negative scalar curvature)
• The Cartan geometry of parabolic type. This is a Cartan geometry modeled on the pair (G,P) where G is a semisimple Lie group and P a parabolic subgroup. Surprisingly, the above mentioned Euclidean geometry is not among these geometries, so this may cause confusion. Simplest examples of parabolic geometries in this sence are projective geometry, conformal geometry and CR geometry.