{"title":"Rolling reductive homogeneous spaces","authors":"Markus Schlarb","doi":"10.1007/s13324-024-00889-z","DOIUrl":null,"url":null,"abstract":"<div><p>Rollings of reductive homogeneous spaces are investigated. More precisely, for a reductive homogeneous space <i>G</i>/<i>H</i> with reductive decomposition <span>\\(\\mathfrak {{g}} = \\mathfrak {{h}} \\oplus \\mathfrak {{m}}\\)</span>, we consider rollings of <span>\\(\\mathfrak {{m}}\\)</span> over <i>G</i>/<i>H</i> without slip and without twist, where <i>G</i>/<i>H</i> is equipped with an invariant covariant derivative. To this end, an intrinsic point of view is taken, meaning that a rolling is a curve in the configuration space <i>Q</i> which is tangent to a certain distribution. By considering an <i>H</i>-principal fiber bundle <span>\\(\\overline{\\pi }:\\overline{Q}\\rightarrow Q\\)</span> over the configuration space equipped with a suitable principal connection, rollings of <span>\\(\\mathfrak {{m}}\\)</span> over <i>G</i>/<i>H</i> can be expressed in terms of horizontally lifted curves on <span>\\(\\overline{Q}\\)</span>. The total space of <span>\\(\\overline{\\pi }:\\overline{Q}\\rightarrow Q\\)</span> is a product of Lie groups. In particular, for a given control curve, this point of view allows for characterizing rollings of <span>\\(\\mathfrak {{m}}\\)</span> over <i>G</i>/<i>H</i> as solutions of an explicit, time-variant ordinary differential equation (ODE) on <span>\\(\\overline{Q}\\)</span>, the so-called kinematic equation. An explicit solution for the associated initial value problem is obtained for rollings with respect to the canonical invariant covariant derivative of first and second kind if the development curve in <i>G</i>/<i>H</i> is the projection of a one-parameter subgroup in <i>G</i>. Lie groups and Stiefel manifolds are discussed as examples.\n</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 2","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s13324-024-00889-z.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis and Mathematical Physics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s13324-024-00889-z","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Rollings of reductive homogeneous spaces are investigated. More precisely, for a reductive homogeneous space G/H with reductive decomposition \(\mathfrak {{g}} = \mathfrak {{h}} \oplus \mathfrak {{m}}\), we consider rollings of \(\mathfrak {{m}}\) over G/H without slip and without twist, where G/H is equipped with an invariant covariant derivative. To this end, an intrinsic point of view is taken, meaning that a rolling is a curve in the configuration space Q which is tangent to a certain distribution. By considering an H-principal fiber bundle \(\overline{\pi }:\overline{Q}\rightarrow Q\) over the configuration space equipped with a suitable principal connection, rollings of \(\mathfrak {{m}}\) over G/H can be expressed in terms of horizontally lifted curves on \(\overline{Q}\). The total space of \(\overline{\pi }:\overline{Q}\rightarrow Q\) is a product of Lie groups. In particular, for a given control curve, this point of view allows for characterizing rollings of \(\mathfrak {{m}}\) over G/H as solutions of an explicit, time-variant ordinary differential equation (ODE) on \(\overline{Q}\), the so-called kinematic equation. An explicit solution for the associated initial value problem is obtained for rollings with respect to the canonical invariant covariant derivative of first and second kind if the development curve in G/H is the projection of a one-parameter subgroup in G. Lie groups and Stiefel manifolds are discussed as examples.
期刊介绍:
Analysis and Mathematical Physics (AMP) publishes current research results as well as selected high-quality survey articles in real, complex, harmonic; and geometric analysis originating and or having applications in mathematical physics. The journal promotes dialog among specialists in these areas.