Smooth Manifolds

This concise and practical textbook presents the essence of the theory on smooth manifolds. A key concept in mathematics, smooth manifolds are ubiquitous: They appear as Riemannian manifolds in differential geometry as space-times in general relativity as phase spaces and energy levels in mechanics as domains of definition of ODEs in dynamical systems as Lie groups in algebra and geometry and in many other areas. The book first presents the language of smooth manifolds, culminating with the Frobenius theorem, before discussing the language of tensors (which includes a presentation of the exterior derivative of differential forms). It then covers Lie groups and Lie algebras, briefly addressing homogeneous manifolds. Integration on manifolds, explanations of Stokes’ theorem and de Rham cohomology, and rudiments of differential topology complete this work. It also includes exercises throughout the text to help readers grasp the theory, as well as more advanced problems for challenge-oriented minds at the end of each chapter. Conceived for a one-semester course on Differentiable Manifolds and Lie Groups, which is offered by many graduate programs worldwide, it is a valuable resource for students and lecturers alike.