一类具有Frobenius双不变度量的\(\mathbf {U_n}\)子群的广义主对数和黎曼性质

Q2 Mathematics

Annali dell''Universita di Ferrara Pub Date : 2024-12-05 DOI:10.1007/s11565-024-00561-1

Donato Pertici, Alberto Dolcetti

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引用次数: 0

摘要

研究了具有自然双不变度量的\(U_n\)的一大闭子群的几何微分性质。对于这些群中的每一个，我们显式地表达了距离函数，直径，最重要的是，我们通过群的李代数中\(P_0^*P_1\)的广义主对数集来参数化具有任意端点\(P_0\)和\(P_1\)的最小测地线段集。证明了这最后一个集合是有限个紧子流形\(\mathfrak {u}_n\)微同构于合适的（明确确定的）齐次空间的非空不相交并。

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Generalized principal logarithms and Riemannian properties of a class of subgroups of \(\mathbf {U_n}\) endowed with the Frobenius bi-invariant metric

We study the geometric-differential properties of a wide class of closed subgroups of \(U_n\) endowed with a natural bi-invariant metric. For each of these groups, we explicitly express the distance function, the diameter, and, above all, we parametrize the set of minimizing geodesic segments with arbitrary endpoints \(P_0\) and \(P_1\) by means of the set of generalized principal logarithms of \(P_0^*P_1\) in the Lie algebra of the group. We prove that this last set is a non-empty disjoint union of a finite number of compact submanifolds of \(\mathfrak {u}_n\) diffeomorphic to suitable (and explicitly determined) homogeneous spaces.

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来源期刊

Annali dell''Universita di Ferrara Mathematics-Mathematics (all)

CiteScore

1.70

自引率

0.00%

发文量

期刊介绍： Annali dell''Università di Ferrara is a general mathematical journal publishing high quality papers in all aspects of pure and applied mathematics. After a quick preliminary examination, potentially acceptable contributions will be judged by appropriate international referees. Original research papers are preferred, but well-written surveys on important subjects are also welcome.