What is "parabolic geometry"

n. geometry based on Euclid's axioms: e.g., only one line can be drawn through a point parallel to another line [syn: elementary geometry, Euclidean geometry]

Parabolic geometry may refer to:

• Euclidean geometry, where Euclidean space is viewed as the natural representation space of the group of Euclidean motions $E(n) = O(n) \ltimes \mathbb{R}^n$
• The geometry of a Riemannian manifold admitting no positive Green's function
• Parabolic geometry (differential geometry): The homogeneous space defined by a semisimple Lie group modulo a parabolic subgroup, or the curved analog of such a space

In differential geometry and the study of Lie groups, a parabolic geometry is a homogeneous space G/P which is the quotient of a semisimple Lie group G by a parabolic subgroup P. More generally, the curved analogs of a parabolic geometry in this sense is also called a parabolic geometry: any geometry that is modeled on such a space by means of a Cartan connection.