赋予双不变度规\(\mathbf {SU_n}\)的一些黎曼性质

IF 1 3区数学 Q1 MATHEMATICS

Annali di Matematica Pura ed Applicata Pub Date : 2024-11-07 DOI:10.1007/s10231-024-01516-1

Donato Pertici, Alberto Dolcetti

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引用次数: 0

摘要

研究了\(SU_n\)上具有Frobenius度规\(\phi \)的一些性质，该度规是\(SU_n\)上唯一的双不变黎曼度规，直到一个正常数倍。特别地，我们用\(P^*Q\)的特征值表示\(P, Q \in SU_n\)之间的距离；我们计算\((SU_n, \phi )\)的直径并确定它的直径对；我们证明了端点为P， Q的所有极小测地线线段的集合可以用紧连通子流形\(\mathfrak {su}_n\)来参数化，该子流形依P和Q微分同构于一个合适的复Grassmannian。

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Some Riemannian properties of \(\mathbf {SU_n}\) endowed with a bi-invariant metric

We study some properties of \(SU_n\) endowed with the Frobenius metric \(\phi \), which is, up to a positive constant multiple, the unique bi-invariant Riemannian metric on \(SU_n\). In particular we express the distance between \(P, Q \in SU_n\) in terms of eigenvalues of \(P^*Q\); we compute the diameter of \((SU_n, \phi )\) and we determine its diametral pairs; we prove that the set of all minimizing geodesic segments with endpoints P, Q can be parametrized by means of a compact connected submanifold of \(\mathfrak {su}_n\), diffeomorphic to a suitable complex Grassmannian depending on P and Q.

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来源期刊

Annali di Matematica Pura ed Applicata 数学-数学

CiteScore

2.10

自引率

10.00%

发文量

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.