Differential geometry on conformal structures and projective structures

在球面上的一条曲线上，由周围球面的共形结构导出一个投影结构。结果表明，如果曲线的投影展开图是内射的，则曲线没有自交。我试着寻找其他空间，而不是球面上的注入性定理。在紧秩对称空间中，只有球具有这种注入性，这是本研究项目的主要成果。这个定理与其他观察结果表明，在所有紧致黎曼流形中，只有球具有这种内射性。这将是一个未来的问题。

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.