Multilinear algebra of finite dimensional vector spaces over fields; differentiable structures and tangent and tensor bundles; differentiable mappings and differentials; exterior differential forms; curves and surfaces as differentiable manifolds; affine connections and covariant differentiation; Riemannian manifolds.
"Lectures on Differential Geometry", Chern, Chen and Lam, World Scientific, 1980.
Syllabus: This course is an introduction to the theory of differential geometry, covering the basic topics like differentiable manifolds, submanifolds, smooth maps, (co)tangent bundles, vector fields, Lie bracket, differential forms, exterior derivative, deRham cohomology, connections, curvatures, Riemannian metrics, etc. More advanced special topics like Riemannian, symplectic, or Poisson geometry may be included if time permits.
(Lerner 2011 )