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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, keeping a constant connection with the basic material. The topics discussed include polar actions, singular Riemannian foliations, cohomogeneity one actions, and positively curved manifolds with many symmetries. This book stems from the experience gathered by the authors in several lectures along the years and was designed to be as self-contained as possible. It is intended for advanced undergraduates, graduate students and young researchers in geometry and can be used for a one-semester course or independent study.

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1: Basic results on Lie groups.- 2: Lie groups with bi-invariant metrics.- 3: Proper and isometric acions.- 4: Adjoint and conjugation actions.- 5: Polar foliations.- 6: Low cohomogeneity actions and positive curvature.- Appendix: Rudiments of smooth manifolds.

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"This book sets out from the geometric point of view the foundations of the theory of Lie groups and Lie algebras, as well as some of the topics relating to the theory of Lie groups of isometric transformations. ... At the end of the book there is an application in which some elements of the theory of smooth manifolds are exposed. The book therefore can be seen as self-contained and thus usable as a textbook." (Vladimir V. Gorbatsevich, Mathematical Reviews, March, 2016)

"The present book provides a nice overview of topics related to isometric actions, exploring relations to active research areas (primarily of the authors), such as isoparametric submanifolds, polar actions, polar foliations, cohomogeneity one actions, and positive curvature via symmetries. ... The book is of great benefit for mature graduate students or researchers in the field." (Andreas Arvanitoyeorgos, zbMATH 1322.22001, 2015)

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Marcos M. Alexandrino is an Associate Professor at the Institute of Mathematics and Statistics of the University of SÆo Paulo, Brazil. He did his PhD at Pontifical Catholic University of Rio de Janeiro, Brazil, with studies at the University of Cologne, in Germany. His research is on the field of Differential Geometry, more specifically on singular Riemannian foliations and isometric actions.

Renato G. Bettiol is a Hans Rademacher Instructor of Mathematics at the University of Pennsylvania, USA. He did his PhD at the University of Notre Dame, USA. His research is on the field of Differential Geometry, more specifically on Riemannian geometry and geometric analysis.

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