This book presents a concise introduction to differential geometry. It is aimed at advanced undergraduate students and first year graduate students who wish to have a basic solid knowledge of the subject, and it can serve as a starting point for more advanced reading. The book is organized into lectures, so it can easily be used as a textbook for a beginning graduate-level course in differential geometry.

Contents:

• 
  

  Basic Concepts:

  
  
  
  
  • Manifolds as Subsets of Euclidean Space
  
  
  • Abstract Manifolds
  
  
  • Manifolds with Boundary
  
  
  • Partitions of Unity
  
  
  • The Tangent Space
  
  
  • The Differential
  
  
  • Immersions, Submersions, and Submanifolds
  
  
  • Embeddings and Whitney's Theorem
  
  
  • Foliations
  
  
  • Quotients
  
  
  
  


• 
  

  Lie Theory:

  
  
  
  
  • Vector Fields and Flows
  
  
  • Lie Bracket and Lie Derivative
  
  
  • Distributions and the Frobenius Theorem
  
  
  • Lie Groups and Lie Algebras
  
  
  • Integrations of Lie Algebras
  
  
  • The Exponential Map
  
  
  • Groups of Transformations
  
  
  
  


• 
  

  Differential Forms:

  
  
  
  
  • Differential Forms and Tensor Fields
  
  
  • Volume Forms and Orientation
  
  
  • Cartan Calculus
  
  
  • Integration on Manifolds
  
  
  • de Rham Cohomology
  
  
  • The de Rham Theorem
  
  
  • Homotopy Invariance and Mayer–Vietoris Sequence
  
  
  • Computations in Cohomology and Applications
  
  
  • The Degree and the Index
  
  
  
  


• 
  

  Fiber Bundles:

  
  
  
  
  • Vector Bundles
  
  
  • The Thom Class and the Euler Class
  
  
  • Pullbacks of Vector Bundles
  
  
  • The Classification of Vector Bundles
  
  
  • Connections and Parallel Transport
  
  
  • Curvature and Holonomy
  
  
  • The Chern–Weil Homomorphism
  
  
  • Characteristic Classes
  
  
  • Fiber Bundles
  
  
  • Principal Fiber Bundles
  
  
  
  

Readership: For advanced undergraduate and beginning graduate level courses in Mathematics/Physics on Differential Geometry.

Key Features:

• The book is much more concise than other textbooks in differential geometry, but it still gives a clear, detailed and careful discussion of the most important topics necessary to grasp the subject


• Some topics, such as the Godement criteria for quotients, are usually not discussed in other textbooks


• Each lecture ends with a list of homework problems, carefully chosen to complete the topics covered in the lecture