Next year I'm doing my second masters year and the teacher in charge of a course called "Differential Geometry" posted online the contents of his class.
I sent him an email asking for the references he used to make his course and never got an answer (I did this with the other courses and they all replied).
Here is the link with the contents of his class ( http://www.mathfds.univ-montp2.fr/files/Affiches/ResumesCours2017-18.pdf ) , it's in French so I'll try to translate it here so you don't have to decipher some math words that may change a lot from English to French.
1) Vector bundle on a manifold, generalities, examples, connection, parallel transport
2) Tangent bundle, riemannian metrics, Levi-Civita connection, covariant derivative
3) Curvature, Riemann tensor, Ricci tensor, sectional curvature, scalar curvature, geometric sense of curvature, constant curvature metrics example, Gauss-Bonnet theorem
4) Variational theory of geodesics, functionals of length and energy, geodesics as extremities of these functionals, hamiltonian system and geodesics, geodesic flow
5) Global properties: exponential map. Riemann distance on manifolds, Hopf-Rinow theorem, Surjectivity of the exponential map, injectivity radius. Example : bi-invariant metrics on compact Lie groups, one-parameter subgroup, example of SL(2;R).
6) Second variation of the length functional, Jacobi fields, critical points of the exponential map, conjugate points, cut locus, examples...
7) Differential forms on a manifold, exterior differential, Cartan formula.
Hoping my translations are somewhat correct, does someone know what his references are for this course ? Or someone knows some good/classic references for differential geometry so I can start to discover the course ?
Thank you for your answers.