Study of conformal differential geometry

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Among many geometric structures of a manifold we are mainly interested in those structures which are closely related to the conformal geometry. Here are some of main results of this research project :1. For a regular curve x : I>M in a conformal manifold M, we can define a projective structure on the interval. Moreover if M is the standard sphere and if the projective developing map of I to the real projective line is injective, then the curve x is injective2. Suppose that we are given a complex number a which is not real and that f is a continuously differentiable complex function defined on a domain in the complex plain. If for four points which have the anharmonic ratio a, the images of the four points also have the same anharmonic ratio, then f is a Moebius transformation. This result is an extension of a theorem by Haruki and Rassias in 1996.3. An analogy of the Yamabe problem in conformal differential geometry is formulated in projective differential geometry. We have given a characterization of the Riemannian connection of an Einstein metric of negative scalar curvature only in terms of affine differential geometry using, the variational method.