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?