非紧凑赫尔墨斯对称空间的荷函数和度量紧凑化

IF 1 3区 数学 Q1 MATHEMATICS
Cho-Ho Chu, María Cueto-Avellaneda, Bas Lemmens
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引用次数: 0

摘要

给定一个非紧凑型的赫米蒂对称空间 M,我们证明,除其他外,M 关于其 Carathéodory 距离的度量紧凑与它的切空间中的闭球同构。我们首先通过把 M 变为一个巴拿赫空间 \((V,\Vert \cdot \Vert )\) 的开单位球 D,并配以一个特殊的约旦结构(称为 \(\textrm{JB}^*\)-triple),给出了对 M 紧凑化中角函数的完整描述。我们识别了 \((V,\Vert \cdot \Vert )\) 度量压缩中的角函数,并通过同构把它的几何和全局拓扑与对偶空间 \(V^*\) 的封闭单位球联系起来。最后,我们证明了在(0\in D\ )处的指数映射(exp _0 :V \longrightarrow D\ )扩展到了((V,\Vert \cdot \Vert ))和((D,\rho ))的度量致密化之间的同构,保留了几何结构,其中(\(\rho \)是 D 上的 Carathéodory 距离)。因此,M 的度量紧凑性可以具体实现为 \((V,\Vert \cdot \Vert )\) 的封闭对偶单位球。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Horofunctions and metric compactification of noncompact Hermitian symmetric spaces

Given a Hermitian symmetric space M of noncompact type, we show, among other things, that the metric compactification of M with respect to its Carathéodory distance is homeomorphic to a closed ball in its tangent space. We first give a complete description of the horofunctions in the compactification of M via the realisation of M as the open unit ball D of a Banach space \((V,\Vert \cdot \Vert )\) equipped with a particular Jordan structure, called a \(\textrm{JB}^*\)-triple. We identify the horofunctions in the metric compactification of \((V,\Vert \cdot \Vert )\) and relate its geometry and global topology, via a homeomorphism, to the closed unit ball of the dual space \(V^*\). Finally, we show that the exponential map \(\exp _0 :V \longrightarrow D\) at \(0\in D\) extends to a homeomorphism between the metric compactifications of \((V,\Vert \cdot \Vert )\) and \((D,\rho )\), preserving the geometric structure, where \(\rho \) is the Carathéodory distance on D. Consequently, the metric compactification of M admits a concrete realisation as the closed dual unit ball of \((V,\Vert \cdot \Vert )\).

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来源期刊
CiteScore
2.10
自引率
10.00%
发文量
99
审稿时长
>12 weeks
期刊介绍: This journal, the oldest scientific periodical in Italy, was originally edited by Barnaba Tortolini and Francesco Brioschi and has appeared since 1850. Nowadays it is managed by a nonprofit organization, the Fondazione Annali di Matematica Pura ed Applicata, c.o. Dipartimento di Matematica "U. Dini", viale Morgagni 67A, 50134 Firenze, Italy, e-mail annali@math.unifi.it). A board of Italian university professors governs the Fondazione and appoints the editors of the journal, whose responsibility it is to supervise the refereeing process. The names of governors and editors appear on the front page of each issue. Their addresses appear in the title pages of each issue.
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