Differential geometry on conformal structures and projective structures

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On a curve in the sphere a projective structure is induced from conformal structure of the ambient sphere. It has been shown that if the projective developing map of the curve is injective the curve has no self-intersection. I tried to find other spaces than the sphere on which the injectivity theorem holds. Among compact rank one symmetric spaces only the sphere has this injectivity property, which is the main result of this research project. This theorem with other observations suggests that among all compact Riemannian manifolds only the sphere may have this injectivity property. This will be a future problem.