Targeted to graduate students of mathematics, this book discusses major topics like the Lie group in the study of smooth manifolds. It is said that mathematics can be learned by solving problems and not only by just reading it. To serve this purpose, this book contains a sufficient number of examples and exercises after each section in every chapter. Some of the exercises are routine ones for the general understanding of topics. The book also contains hints to difficult exercises. Answers to all exercises are given at the end of each section. It also provides proofs of all theorems in a lucid manner. The only pre-requisites are good working knowledge of point-set topology and linear algebra.
Author(s): Manjusha Majumdar, Arindam Bhattacharyya
Series: University Texts in the Mathematical Sciences
Publisher: Springer
Year: 2023
Language: English
Pages: 218
City: Singapore
Preface
Contents
About the Authors
List of Figures
1 Calculus on mathbbRn
1.1 Smooth Functions
1.2 Tangent Vector
1.3 Germ of a Function
1.4 Inverse Function Theorem
1.5 Implicit Function Theorem
2 Manifold Theory
2.1 Topological Manifold
2.2 Smooth Germs on a Topological Manifold
2.3 Smooth Manifold
2.4 Stereographic Projection
2.5 Orientable Surface
2.6 Product Manifold
2.7 Smooth Function on Smooth Manifold
2.8 Differential Curve and Tangent Vector
2.9 Inverse Function Theorem for Smooth Manifold
2.10 Vector Field
2.11 Integral Curve
2.12 Differential of a Mapping
2.13 Submanifolds
2.14 f-Related Vector Fields
2.15 One Parameter Group of Transformations on a Manifold
3 Differential Forms
3.1 Cotangent Space
3.2 r-form, Exterior Product
3.3 Exterior Differentiation
3.4 Pull-Back Differential Form
4 Lie Group
4.1 Lie Group, Left and Right Translation
4.2 Invariant Vector Field
4.3 Invariant Differential Form
4.4 Automorphism
4.5 One-Parameter Subgroup of a Lie Group
4.6 Lie Transformation Group (Action of a Lie Group on a Manifold)
Appendix References
