The First Adventures on Differential Geometry

This book serves as a friendly stepping stone for readers who wish to start learning differential geometry. It lays down the most essential foundations, ensuring that readers feel prepared to tackle more advanced concepts in the future. Differential geometry is often perceived or presented as a myriad of seemingly disjointed ideas to beginners. This book bridges those gaps by following the story of a character named Beng, along with his friends, as they journey to master differential geometry. Starting with the basics in Euclidean space, the book offers an accessible entry point into understanding geometry. It then progresses to curved surfaces and intrinsic geometry, culminating in a quintessential application of Ricci curvature. Readers will explore the Einstein equation and work through the Schwarzschild solution, ultimately calculating the event horizon for a non-rotating black hole. The exercises throughout the book are crafted to develop a deep understanding of the subject, with detailed solutions provided to guide readers through each step. Based on a series of lecture notes from a course designed for a broad audience, this book has been classroom-tested, proving its effectiveness as an introductory and illuminating textbook. Contents: Differentials on Surfaces Mathematical Preliminaries Fields, Derivatives and Curvilinear Coordinates Introduction to Curved Surfaces Embedded in ℝn Covariant Derivatives and Parallel Transport on M ⊂ ℝn Intrinsic Geometry The Lie Derivative Riemann Curvature Tensor Putting It All Together Readership: Undergraduates, advanced undergraduates and researchers studying or applying differential geometry within their fields of interest. Researchers from different disciplines may also learn to formulate open problems with geometry, including research in bioinformatics, machine learning, artificial intelligence, robotics, applied mathematics, economics, social sciences and clinical sciences. Key Features: Materials presented are accessible at the level of freshman mathematics Offers only the most essential components of differential geometry to give readers a good foundation to study the subject at a more advanced level Exercises, with detailed solutions, are designed to guide readers towards understanding the subject This book is classroom-tested, based on a differential geometry course for a broad audience