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added this quote to before the Idea-section:
In the wake of the movement of ideas which followed the general theory of relativity, I was led to introduce the notion of new geometries, more general than Riemannian geometry, and playing with respect to the different Klein geometries the same role as the Riemannian geometries play with respect to Euclidean space. The vast synthesis that I realized in this way depends of course on the ideas of Klein formulated in his celebrated Erlangen programme while at the same time going far beyond it since it includes Riemannian geometry, which had formed a completely isolated branch of geometry, within the compass of a very general scheme in which the notion of group still plays a fundamental role.
[Élie Cartan 1939, as quoted in Sharpe 1997, p. 171]
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as an example, I have spelled out (here) the “round $S^3$” as a Cartan geometry
(This example would fit into other entries, too, such as at first-order formulation of gravity. Indeed, I wrote this to sort out the sign convention for the scalar curvature in the example discussed at Freund-Rubin compactification – Details.)
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Élie Cartan, La géométrie des espaces de Riemann, Mémorial des sciences mathématiques 9 (1925) [numdam:MSM_1925__9__1_0]
Élie Cartan (translated by Vladislav Goldberg from Cartan’s lectures at the Sorbonne in 1926–27): Riemannian Geometry in an Orthogonal Frame, World Scientific (2001) [doi:10.1142/4808, pdf]
Élie Cartan (translated by Robert Hermann from Cartan’s lectures in 1951): Geometry of Riemannian Spaces, Lie Groups: History, Frontiers and Applications XIII, Math Sci Press (1983) [ark:/13960/s28rzmj9xrv]
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