Contents

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

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