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As far as I am aware, elliptic geometry refers to a geometry which satisfies all of the axioms of Euclidean geometry, except that the parallel postulate (given a line and a point not on that line, there is exactly one parallel line through that point) does not hold and is replaced with the postulate that, given a line and point not on that line, there is no line through that point which is parallel.

Since any two lines in projective space intersect, does this make projective geometry a special case of elliptic geometry? I know that spherical geometry is a special case of elliptic geometry, and that any two lines intersect in two points in spherical geometry, as opposed to only one point in projective geometry -- still, both of these seem to satisfy the "elliptic parallel postulate".

Am I misunderstanding this?

Also, the complex projective line $\mathbb{CP}^1$ is a Riemann sphere, so does this make spherical geometry somehow realizable as a special case of projective geometry?

Is elliptic geometry just the geometry of any space with positive curvature, because this curvature causes the parallel postulate to fail, and thus can my question just be reduced to whether projective spaces (real and/or complex) have positive curvature?

EDIT: On Wikipedia it says that projective space is a model for elliptic geometry, and that projective space is "an abstract elliptic geometry". So apparently projective geometry is a sub-case of (less general than) elliptic geometry?

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    $\begingroup$ One of the assumptions that Euclid makes, that is not in his axioms, is that if you have three distinct points on a line, then exactly one of those points is "between" the other two. This is violated in both Projective geometry and spherical geometry. But it doesn't technically violate Euclid's postulates. $\endgroup$
    – Thomas Andrews
    Oct 15, 2016 at 13:35
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    $\begingroup$ As for $\mathbb CP^1$ and the sphere. As a geometry, $\mathbb CP^1$ is a one-dimensional space - a line. The sphere can be given a higher-dimensional geometry, but it violates the axiom that two points determine exactly one line. (Pick two points on opposite sides of the circle.) $\endgroup$
    – Thomas Andrews
    Oct 15, 2016 at 13:38
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    $\begingroup$ @william if I remember correctly, identifying diametrically opposite points on the sphere with each other restores the axiom that lines intersect in exactly one point. At that point, you are basically using the model of projective geometry where points are lines through the origin. But 'spherical geometry' seems to apply only to the sphere with lines intersecting in two points. $\endgroup$
    – rschwieb
    Oct 15, 2016 at 19:09

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Projective geometry is the geometry of incidence. Elliptic geometry is a metric geometry. Euclidean, hyperbolic and elliptic geometry can all be derived from projective geometry by choosing an appropriate involution and restricting the domain of points. Projective geometry arose historically, by adding ideal points to the Euclidean plane, so all of its theorems have their Euclidean analogs.

This is just (a small) part of the whole story. But I hope it helps.

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