Study of conformal differential geometry

在流形的众多几何结构中，我们主要对那些与共形几何密切相关的结构感兴趣。1.对于共形流形M中的正则曲线x：i&gt；M，我们可以在区间上定义一个射影结构。此外，如果M是标准球面，且如果I到实射影线的射影展开映射是内射的，则曲线x是内射的。假设我们得到一个复数a，它不是实数，f是定义在复平面上的一个区域上的连续可微复函数。如果对具有非谐比a的四个点，这四个点的像也具有相同的非谐比，则f是Moebius变换。这一结果是Haruki和Rassias在1996.3中的一个定理的推广。在射影微分几何中建立了共形微分几何中Yamabe问题的一个类比。我们仅用仿射微分几何的方法，用变分方法给出了负标量曲率的爱因斯坦度量的黎曼联络的一个刻画。

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.