Lie Groups and Geometric Aspects of Isometric Actions

✍ Scribed by Marcos M. Alexandrino, Renato G. Bettiol

• Publisher: Springer
• Year: 2015
• Tongue: English
• Leaves: 215
• Edition: 2015
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Preface
Contents
Part I Lie Groups
1 Basic Results on Lie Groups
1.1 Lie Groups and Lie Algebras
1.2 Lie Subgroups and Lie Homomorphisms
1.3 Exponential Map and Adjoint Representation
1.4 Closed Subgroups and More Examples
2 Lie Groups with Bi-invariant Metrics
2.1 Basic Facts of Riemannian Geometry
2.2 Bi-invariant Metrics
2.3 Killing Form and Semisimple Lie Algebras
2.4 Splitting Lie Groups with Bi-invariant Metrics
Part II Isometric Actions
3 Proper and Isometric Actions
3.1 Proper Actions and Fiber Bundles
3.2 Slices and Tubular Neighborhoods
3.3 Isometric Actions
3.4 Principal Orbits
3.5 Orbit Types
4 Adjoint and Conjugation Actions
4.1 Maximal Tori and Polar Actions
4.2 Normal Slices of Conjugation Actions
4.3 Roots of a Compact Lie Group
4.4 Weyl Group
4.5 Dynkin Diagrams
5 Polar Foliations
5.1 Definitions and First Examples
5.2 Holonomy and Orbifolds
5.3 Surgery and Suspension of Homomorphisms
5.4 Differential and Geometric Aspects of Polar Foliations
5.5 Transnormal and Isoparametric Maps
5.6 Perspectives
6 Low Cohomogeneity Actions and Positive Curvature
6.1 Cheeger Deformation
6.2 Compact Homogeneous Spaces
6.3 Cohomogeneity One Actions
6.4 Positive and Nonnegative Curvature via Symmetries
AppendixA Rudiments of Smooth Manifolds
A.1 Smooth Manifolds
A.2 Vector Fields
A.3 Foliations and the Frobenius Theorem
A.4 Differential Forms, Integration, and de Rham Cohomology
References
Index