Multivariable calculus, basic notions in topology
Vector fields and and Lie derivatives
Tensors and differential forms
Exterior algebras and derivatives
Metric and curvature
Moving frames and structure equations
Time permitting, Hodge's Theory
Educational goals: the aim of the course is to use calculus tools on "curved objects", and to relate local/differential concepts to global ones, with applications to the study of differentiable and Riemannian manifolds.
M. Spivak, A comprehensive introduction to Differential Geometry, Publish or Perish.
R. Abraham, J. Marsden, T. Ratiu, Manifolds, tensor analysis, and applications, Springer.
F. Warner, Foundations of differentiable manifolds and Lie groups, Springer.