Lie Groups and Geometric Aspects of Isometric Actions (eBook)

Marcos M. Alexandrino is an Associate Professor at the Institute of Mathematics and Statistics of the University of São 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.

Preface 8
Contents 10
Part I Lie Groups 12
1 Basic Results on Lie Groups 13
1.1 Lie Groups and Lie Algebras 13
1.2 Lie Subgroups and Lie Homomorphisms 17
1.3 Exponential Map and Adjoint Representation 23
1.4 Closed Subgroups and More Examples 28
2 Lie Groups with Bi-invariant Metrics 36
2.1 Basic Facts of Riemannian Geometry 36
2.2 Bi-invariant Metrics 47
2.3 Killing Form and Semisimple Lie Algebras 50
2.4 Splitting Lie Groups with Bi-invariant Metrics 54
Part II Isometric Actions 57
3 Proper and Isometric Actions 58
3.1 Proper Actions and Fiber Bundles 58
3.2 Slices and Tubular Neighborhoods 71
3.3 Isometric Actions 76
3.4 Principal Orbits 80
3.5 Orbit Types 83
4 Adjoint and Conjugation Actions 92
4.1 Maximal Tori and Polar Actions 92
4.2 Normal Slices of Conjugation Actions 99
4.3 Roots of a Compact Lie Group 100
4.4 Weyl Group 106
4.5 Dynkin Diagrams 109
5 Polar Foliations 115
5.1 Definitions and First Examples 115
5.2 Holonomy and Orbifolds 117
5.3 Surgery and Suspension of Homomorphisms 122
5.4 Differential and Geometric Aspects of Polar Foliations 123
5.5 Transnormal and Isoparametric Maps 132
5.6 Perspectives 141
6 Low Cohomogeneity Actions and Positive Curvature 144
6.1 Cheeger Deformation 144
6.2 Compact Homogeneous Spaces 150
6.3 Cohomogeneity One Actions 162
6.4 Positive and Nonnegative Curvature via Symmetries 176
AppendixA Rudiments of Smooth Manifolds 189
A.1 Smooth Manifolds 189
A.2 Vector Fields 191
A.3 Foliations and the Frobenius Theorem 194
A.4 Differential Forms, Integration, and de Rham Cohomology 196
References 202
Index 211