Smooth Manifolds

1. Front Matter

   Pages i-xii

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2. Smooth Manifolds

   • Claudio Gorodski

   Pages 1-45

3. Tensor Fields and Differential Forms

   • Claudio Gorodski

   Pages 47-70

4. Lie Groups

   • Claudio Gorodski

   Pages 71-99

5. Integration

   • Claudio Gorodski

   Pages 101-137

6. Back Matter

   Pages 139-154

   PDF

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. 

• smooth manifold
• Lie groups
• differential topology
• Tensor analysis
• differential geometry

“The work is written in a clear and precise style. The notions are very well presented and many examples are given. Moreover, at the end of each chapter, there are several challenging problems for gifted students. In the reviewer’s opinion, this monograph will be of great interest to graduate students and researchers working in the field of differential geometry.” (Gabriel Eduard Vilcu, zbMATH 07235511, 2020)

• Institute of Mathematics and Statistics, University of São Paulo, São Paulo, Brazil

  Claudio Gorodski

Claudio Gorodski is a Full Professor at the Institute of Mathematics and Statistics, University of São Paulo, Brazil. He holds a PhD in Mathematics (1992) from the University of California at Berkeley, USA, and a Habilitation degree (1998) from the University of São Paulo, Brazil. His research interests include Lie transformation groups in Riemannian geometry, geometry of submanifolds, Riemannian symmetric spaces, and sub-Riemannian geometry.

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