A review of continuity and differentiability in more than one variable. The inverse, implicit, and constant rank theorems.
Definitions and examples of manifolds, maps between manifolds, regular and critical values, partition of unity, Sard’s theorem and applications.
Tangent spaces and the tangent/cotangent bundles, definition of general vector bundles, vector fields and flows, Frobenius’ theorem.
Tensors, differential forms, Lie derivative and the exterior derivative, integration on manifolds, Stokes’ theorem.
Introduction to de Rham cohomology.
Suggested books and references:
Tu, Loren, An Introduction to Manifolds, Universitext, Springer-Verlag 2011.
John Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, Springer-Verlag 2012.
Barden, Dennis and Thomas, Charles, An Introduction to Differential Manifolds, World Scientific 2003.
Spivak, Michael, Comprehensive Introduction to Differential Geometry, Vol 1, Publish or Perish, 2005.