An Introduction to Smooth Manifolds

Preface

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 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

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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 . . . . . . . . . . . . . . . . . . . . . . 912.9 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

MANJUSHA MAJUMDAR (TARAFDAR), PhD, former Professor of Pure Mathematics at the University of Calcutta, India. Her main interest lies in smooth manifolds. She has published a number of research papers in several international journals of repute. Under her supervision, 10 students have already been awarded their PhD degrees. She is member of many national and international mathematical societies and serves on the editorial board of several journals of repute. She has visited several institutions in India and abroad on invitation. She is the Principal Investigator of the prestigious e-PG Pathshala in Mathematics and MOOC in Mathematics, recommended by the Ministry of Human Resource Development, the Government of India. She co-authored a book entitled, Differential Geometry.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.