Curvature and the equivalence problem in sub-Riemannian geometry

IF 0.5 Q3 MATHEMATICS
E. Grong
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引用次数: 3

Abstract

These notes give an introduction to the equivalence problem of sub-Riemannian manifolds. We first introduce preliminaries in terms of connections, frame bundles and sub-Riemannian geometry. Then we arrive to the main aim of these notes, which is to give the description of the canonical grading and connection existing on sub-Riemann manifolds with constant symbol. These structures are exactly what is needed in order to determine if two manifolds are isometric. We give three concrete examples, which are Engel (2,3,4)-manifolds, contact manifolds and Cartan (2,3,5)-manifolds. These notes are an edited version of a lecture series given at the 42nd Winter school: Geometry and Physics, Snrí, Check Republic, mostly based on [Gro20] and other earlier work. However, the work on Engel (2,3,4)-manifolds is original research, and illustrate the important special case were our model has the minimal set of isometries.
次黎曼几何中的曲率与等价问题
这些笔记介绍了次黎曼流形的等价问题。我们首先从连接、框架束和亚黎曼几何的角度进行初步介绍。然后我们到达了这些笔记的主要目的,即给出具有常数符号的亚黎曼流形上存在的正则级配和连接的描述。这些结构正是确定两个流形是否是等距的所需要的。我们给出了Engel(2,3,4)-流形、接触流形和Cartan(2,3,5)-流形三个具体的例子。这些笔记是第42届冬季学校系列讲座的编辑版本:几何和物理,Snrí, Check Republic,主要基于[Gro20]和其他早期工作。然而,关于Engel(2,3,4)-流形的工作是原创的研究,并且说明了我们的模型具有最小等距集的重要特殊情况。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Archivum Mathematicum
Archivum Mathematicum MATHEMATICS-
CiteScore
0.70
自引率
16.70%
发文量
0
审稿时长
35 weeks
期刊介绍: Archivum Mathematicum is a mathematical journal which publishes exclusively scientific mathematical papers. The journal, founded in 1965, is published by the Department of Mathematics and Statistics of the Faculty of Science of Masaryk University. A review of each published paper appears in Mathematical Reviews and also in Zentralblatt für Mathematik. The journal is indexed by Scopus.
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