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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…mehr
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- Produktdetails
- Verlag: Springer Nature Singapore
- Seitenzahl: 210
- Erscheinungstermin: 1. Juni 2023
- Englisch
- ISBN-13: 9789819905652
- Artikelnr.: 68159763
- Verlag: Springer Nature Singapore
- Seitenzahl: 210
- Erscheinungstermin: 1. Juni 2023
- Englisch
- ISBN-13: 9789819905652
- Artikelnr.: 68159763
ARINDAM BHATTACHARYYA, PhD, is Professor, Department of Mathematics, Jadavpur University, Kolkata, India. His main interest lies in smooth manifolds, geometric flows and computational geometry. He has published a number of research papers in several international journals of repute. Under his supervision, 19 research scholars have already been awarded their PhD degrees and 8 are pursuing their research. He is member of many national and international mathematical societies and serves on the editorial board of several journals of repute. He has visited several institutions in India and abroad on invitation. He has organized several international conferences and research schools in collaboration with internationally reputed mathematical organizations. He is the Course Coordinator of Differential Geometry of the prestigious e-PG Pathshala in Mathematics and MOOC in Mathematics, recommended by the Ministry of Human Resource Development, the Government of India and of Netaji Subhas Open University, India. He co-authored a book entitled, Differential Geometry with Prof. Manjusha Majumdar.
1 Calculus on Rn ..........................5
1.1 Smooth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Tangent Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Germ of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 Manifold Theory 47
2.1 Topological Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 Smooth Germs on a topological manifold . . . . . . . . . . . . . . . . . . . 55
2.3 Smooth Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4 Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
iii
iv CONTENTS
2.5 Orientable Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.6 Product Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.7 Smooth Function on Smooth Manifold . . . . . . . . . . . . . . . . . . . . 84
2.8 Differential Curve, Tangent Vector . . . . . . . . . . . . . . . . . . . . . . 91
2.9 Inverse Function Theorem for Smooth Manifold . . . . . . . . . . . . . . . 97
2.10 Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2.11 Integral Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.12 Differential of a Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2.13 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
2.14 f-related vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
2.15 One Parameter Group of Transformations on a Manifold . . . . . . . . . . 149
1 Calculus on Rn ..........................5
1.1 Smooth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Tangent Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Germ of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 Manifold Theory 47
2.1 Topological Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 Smooth Germs on a topological manifold . . . . . . . . . . . . . . . . . . . 55
2.3 Smooth Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4 Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
iii
iv CONTENTS
2.5 Orientable Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.6 Product Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.7 Smooth Function on Smooth Manifold . . . . . . . . . . . . . . . . . . . . 84
2.8 Differential Curve, Tangent Vector . . . . . . . . . . . . . . . . . . . . . . 91
2.9 Inverse Function Theorem for Smooth Manifold . . . . . . . . . . . . . . . 97
2.10 Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2.11 Integral Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.12 Differential of a Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2.13 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
2.14 f-related vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
2.15 One Parameter Group of Transformations on a Manifold . . . . . . . . . . 149