Half-order differentials on Riemann surfaces
Half-order differentials on Riemann surfaces
复制标题
黎曼曲面上的半阶微分
DOI：
10.1137/0114073
复制
发表时间：
1966
期刊：
影响因子：
--
通讯作者：
M. Schiffer
中科院分区：
文献类型：
--
作者：
N. Hawley;M. Schiffer
在本文中，我们希望展示半整数阶微分在黎曼曲面理论中的效用。我们已经发现，89阶和89阶的微分隐式地涉及到许多早期的研究中，例如，庞加莱对傅氏函数和黎曼曲面上的微分方程的研究。但是，明确承认这些差异是值得研究的实体，其本身的价值似乎是新的。我们相信这样的研究将对Riemarm曲面理论的各个方面产生相当大的统一作用，我们希望通过实例和应用来展示如何通过研究这些半阶微分来澄清和整合该理论的某些部分。处理半阶微分的一个强大的潜在原因来自于一般的轮廓积分技术；黎曼已经介绍过了。在标准理论中，人们对一个微分（线性）对一个阿贝尔积分（加性函数）积分，并使用周期关系和剩余定理得到一个t恒等式。正如我们将要演示的，我们可以做一个类似的事情，将两个89阶的微分相乘，并使用相同的轮廓积分技术。正如经常发生的那样，当一个人发现了一个新的（至少对他来说）实体，并开始环顾四周，看看它在哪里自然地出现时，他会惊讶地发现它的许多藏身之处，而且都离表面很近。我们目前关于黎曼曲面研究的观点是从早期的观点演变而来的，在早期的观点中，我们引入了亚纯连接的概念，类比于实微分几何中的经典概念；我们现在来看连接理论
In this paper we wish to exhibit the utility of differentials of half integer order in the theory of Riemann surfaces. We have found tha t differentials of order 89 and order 8 9 have been involved implicitly in numerous earlier investigations, e.g., Poincar~'s work on Fuchsian functions and differential equations on Riemann surfaces. But the explicit recognition of these differentials as entities to be studied for their own worth seems to be new. We believe tha t such a s tudy will have a considerable unifying effect on various aspects of the theory of Riemarm surfaces, and we wish to show, by means of examples and applications, how some parts of this theory are clarified and brought together through investigating these half-order differentials. A strong underlying reason for dealing with half-order differentials comes from the general technique of contour integration; already introduced by Riemann. In the standard theory one integrates a differential (linear) against an Abelian integral (additive function) and uses period relations and the residue theorem to arrive a t identities. As we shall demonstrate, one can do an analogous thing by multiplying two differentials of order 89 and using the same techniques of contour integration. As often happens, when one discovers a new (at least to him) enti ty and starts looking around to see where it occurs naturally, one is stunned to find so many of its hiding places a n d all so near the surface. Our current point of view concerning the s tudy of Riemann surfaces has evolved from an earlier one in which we introduced the notion of a meromorphic connection in analogy with classical notions in real differential geometry; we now view the theory of connections