Lie Groups and Geometric Aspects of Isometric Actions
This book provides quick access to the theory of Lie groups and isometric actions on smooth manifolds, using a concise geometric approach. After a gentle introduction to the subject, some of its recent applications to active research areas are explored, k
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e Groups and Geometric Aspects of Isometric Actions
Lie Groups and Geometric Aspects of Isometric Actions
Marcos M. Alexandrino • Renato G. Bettiol
Lie Groups and Geometric Aspects of Isometric Actions
123
Marcos M. Alexandrino Departamento de Matemática Instituto de Matemática e Estatística Universidade de São Paulo São Paulo, Brazil
Renato G. Bettiol Department of Mathematics University of Pennsylvania Philadelphia, PA, USA
ISBN 978-3-319-16612-4 ISBN 978-3-319-16613-1 (eBook) DOI 10.1007/978-3-319-16613-1 Library of Congress Control Number: 2015936906 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)
Dedicated to our families.
Preface
This book is intended for advanced undergraduates, graduate students, and young researchers in geometry. It was written with two main goals in mind. First, we give a gentle introduction to the classical theory of Lie groups, using a concise geometric approach. Second, we provide an overview of topics related to isometric actions, exploring their relations with the research areas of the authors and giving the main ideas of proofs. We discuss recent applications to active research areas, such as isoparametric submanifolds, polar actions and polar foliations, cohomogeneity one actions, and positive curvature via symmetries. In this way, the text is naturally divided in two interrelated parts. Let us give a more precise description of such parts. The goal of the first part (Chaps. 1 and 2) is to introduce the concepts of Lie groups, Lie algebras and adjoint representation, relating these objects. Moreover, we give basic results on closed subgroups, bi-invariant metrics, Killing forms, and splitting of Lie algebras in simple ideals. This is done concisely due to the use of Riemannian geometry, who
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