A characterization of bounded symmetric domains by curvature
A characterization of bounded symmetric domains by curvature
复制标题
通过曲率表征有界对称域
DOI：
10.1090/s0002-9947-1983-0688960-x
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发表时间：
1983
影响因子：
1.3
通讯作者：
Dotti Miatello
中科院分区：
文献类型：
--
作者：
BY Curvature;Dotti Miatello
本文将证明一个有界齐性区域是对称的当且仅当在Bergman度量下，所有的截面曲率都是非正的。导论.有界对称整环在Bergman度量下具有非正的截面曲率。本文讨论了有界齐性域（等价于齐性Siegel域）在Bergman度量下具有非正截面曲率时，必是对称域的逆问题.证明使用了正规j-代数[1-3]的技巧，以及第一作者Vinberg [16，17]和间接Dorfmeister [5，6]的结果。本文的主体分为两个部分。首先，我们导出了不可约正规j-代数的根空间的维数之间的某些关系，在附加曲率假设存在的情况下，这些关系证明了对应于某些根的所有根空间的维数相等，对于某些根，没有倍数是根，而对应于另一根的一半的根的所有根空间的维数相等。在第二节中，我们证明了第一维结果意味着相应的Siegel域的锥是自对偶的，而二维结果一起意味着该域是拟对称的。然后证明完成，因为[3]在Bergman度量中具有非正截面曲率的拟对称域已知是对称的。我们注意到，上面提到的一些附属结果本质上并不新鲜。然而，为了表达的一致性，有必要用正规jalgebras的语言来表达它们，因此将它们包括在内。这里也与泽洛（Lundquist）的工作重叠。1.在本文中，（s，j）表示具有容许形式w）的正规j-代数。这意味着s是有限维真实的分裂可解李代数，具有几乎复结构j：s -）s使得[X，Y] + j[jX，Y]+ j[X，jY] = [jX，jY]并且w是s上的线性形式使得双线性形式= o][jX，Y]是对称的、正定的，和j-不变量（注意，在[2]中错误地省略了s是真实的分裂的假设）。设n = [s，s]，a是n在s中的正交补.根据Pyatetskiii-Shapiro [13]的基本结构定理，a是一个可交换的。1980年数学学科分类。初级32 MI 0、53 C30。1983年美国数学学会？0002-9947/82/000-0759/$09.00
This paper will prove that a bounded homogeneous domain is symmetric if and only if, in the Bergman metric, all sectional curvatures are nonpositive. Introduction. It is well known that a bounded symmetric domain has nonpositive sectional curvature in the Bergman metric. This paper is devoted to the converse, namely, that a bounded homogeneous domain (equivalently, homogeneous Siegel domain) which has nonpositive sectional curvature in the Bergman metric must be a symmetric domain. The proof uses the techniques of normal j-algebras [13], as well as results of Vinberg [16, 17], the first author [1-3], and, indirectly, Dorfmeister [5, 6]. The body of the paper is in two sections. In the first, we derive certain relations between the dimensions of the root spaces in an irreducible normal j-algebra, which, in the presence of the additional curvature assumption, show the equality of the dimensions of all root spaces corresponding to certain roots for which no multiple is a root and the equality of the dimensions of all root spaces corresponding to roots which are half of another root. In the second section, we show that the first dimension result implies that the cone of the corresponding Siegel domain is self-dual while the two dimension results together imply that the domain is quasi-symmetric. The proof is then finished because [3] a quasi-symmetric domain with nonpositive sectional curvature in the Bergman metric is known to be symmetric. We remark that some of the subsidiary results mentioned above are not essentially new. However, for consistency of presentation, it is necessary to have them in the language of normal jalgebras and for that reason they are included. There is also overlap here with work of Zelow (Lundquist) [21]. 1. Throughout this paper, (s, j) will denote a normal j-algebra with admissible form w). This means that s is a finite dimensional real split solvable Lie algebra with almost complex structure j: s -) s such that [X, Y] + j[jX, Y] + j[X, jY] = [jX, jY] and w is a linear form on s such that the bilinear form = o][jX, Y] is symmetric, positive-definite, and j-invariant (note that the assumption that s is real split was incorrectly omitted in [2]). Let n = [s, s] and let a be the orthogonal complement of n in s. By the basic structure theorem of Pyatetskii-Shapiro [13], a is a commutative Received by the editors July 13, 1981. 1980 Mathematics Subiect Classification. Primary 32MI0, 53C30. ' 1983 American Mathematical Society ? 0002-9947/82/000-0759/$09.00