A Course in Riemannian Geometry in pdf

Contents

• Smooth Manifolds 
  • Smooth Manifolds, Submanifolds 
  • Smooth Mappings between Smooth Manifolds 
  • Bump Functions and Partitions of Unity 
• Tangent Spaces
  • Derivatives of Smooth Maps 
  • Vector Fields on Smooth Manifolds, The Lie Bracket
• Affine Connections on Smooth Manifolds
  • Vector Fields along Smooth Maps
  • Covariant Differentiation of Vector Fields along Curves
  • Vector Fields along Parameterized Surfaces
• Riemannian Manifolds
  • The Levi-Civita Connection
• Geometry of Surfaces in R 3
• Geodesics in Riemannian Manifolds
  • Length-Minimizing Curves in Riemannian Manifolds 
  • Geodesic Spheres and Gauss’ Lemma
• Complete Riemannian Manifolds
  • Local Isometries and Covering Maps
• Jacobi Fields
  • Flat Riemannian manifolds
  • The Cartan-Hadamard Theorem, The Second Variation of Energy