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Riemann was the first to talk about elliptic geometry: Bolyai and Lobačevskij (even Gauss too) studied only hyperbolic geometry. But of course some theorems of (planar) elliptic geometry were known since more than fifteen centuries, since the surface of a sphere is a space of constant positive curvature even if it's not a real model of elliptic geometry.

Did Riemann actually notice this? If not, who was the first to point out the equivalence?

[EDIT: I tried to explain what I meant. The author of the approved answer read anyway into my mind!]

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  • $\begingroup$ A sphere/circle is an ellipse. The foci just happen to be the same. So what's the real question here/ $\endgroup$ – Carl Witthoft Apr 2 '18 at 13:30
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Elliptic geometry is not equivalent to geometry on the sphere because there is non-unique line through antipodal points on the sphere, contrary to one of the axioms. One needs to identify the antipodal points, which gives the real projective plane. With induced metric it will be a model.

According to Kline, Riemann introduced manifolds of constant curvature in his 1861 paper, although there is some controversy as to what can be read into it. It is also unclear if he was aware that two different geometries were involved for constant positive curvature, it was later explicitly pointed out by Klein. Beltrami, who also showed that the pseudo-sphere is a (partial) model of hyperbolic geometry, pointed out that the sphere is a model for one of them. See Jim Murdock's post on Math Forum for relevant quotes from Kline's Mathematical Thought from Ancient to Modern Times, the controversy is also discussed there.

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