Nonholonomic geometry on finite and infinite dimensional Lie groups and rolling manifolds
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Part I: We start by giving some background on the topics discussed in this thesis. The main topic of the thesis is nonholonomic geometry. In Chapter 1 we give an introduction of nonholonomic geometry in the context of geometric control theory. In a brief exposition, we try to give an overview of the areas of sub-Riemannian and sub-Lorentzian geometry, stating several of the most important results in this area. A historical account concludes this chapter. Chapters 2 and 3 consist of mathematical prerequisits for the later presented results. However, these chapters mainly focus on certain selected facts, rather than trying to give an overview of a whole topic. Chapter 2 contains some results from differential geometry related to submersions and geodesic curvatures. Chapter 3 gives introductory remarks on the convenient calculus of infinite dimensional manifolds. Chapter 4, the last chapter in part I, gives a short presentation and summary of the main results of the papers included in Part II. We first present the results of Paper B, regarding sub-Riemannian and sub-Lorentzian geometry on the universal cover of SU(1, 1). The results in Papers C, D and F are then considered, which concern the nonholonomic dynamical system of two manifolds rolling on each other without twisting or slipping. Finally, we present some results in infinite dimensional manifolds in Paper A and Paper F. In particular, Paper F contains a generalization of sub-Riemannian geometry to the infinite dimensional setting. Part I ends with the bibliography of the 4 first chapters. Part II: Here, six papers are included, Papers A to F. Papers are listed in chronological order according to their date of completion. Two of them are published, one is accepted for publication and three are submitted.
Paper I: Erlend Grong, Pavel Gumenyuk and Alexander Vasil'ev. Matching univalent functions and conformal welding. Annales Academiæ Scientiarum Fennicæ Mathematica 34(1): 303 – 314, 2009. Full text not available in BORA due to publisher restrictions. The article is available at: http://www.acadsci.fi/mathematica/Vol34/GrongGumenyukVasilev.html Pre-print version available at: http://arxiv.org/abs/0806.0930Paper II: Erlend Grong and Alexander Vasil'ev. Sub-Riemannian and sub-Lorentzian geometry on SU(1; 1) and on its universal cover. Journal of Geometric Mechanics volume 3 (2): 225 – 260, July 2011. Full text not available in BORA due to publisher restrictions. The article is available at: http://dx.doi.org/10.3934/jgm.2011.3.225 Pre-print version available at: http://arxiv.org/abs/0910.0945Paper III: Mauricio Godoy Molina, Erlend Grong, Irina Marika and Fátima Silva Leite. An intrinsic formulation of the rolling manifolds problem. Journal of Dynamical and Control Systems 18(2): 181–214, April 2012. Full text not available in BORA due to publisher restrictions. The article is available at: http://dx.doi.org/10.1007/s10883-012-9139-2 Pre-print version available at: http://arxiv.org/abs/1008.1856Paper IV: Erlend Grong. Controllability of rolling without twisting or slipping in higher dimensions. Full text not available in BORA. Pre-print version available at: http://arxiv.org/abs/1103.5258Paper V: Mauricio Godoy Molina and Erlend Grong. Geometric condition for the existence of an intrinsic rolling. Full text not available in BORA. Pre-print version available at: http://arxiv.org/abs/1111.0752Paper VI: Erlend Grong, Irina Markina and Alexander Vasil'ev. Infinite-dimensional sub-Riemannian geometry. Full text not available in BORA. Pre-print version available at: http://arxiv.org/abs/1201.2251