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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.

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    $\begingroup$ A professor (Jack Lee) at my school wrote: sites.math.washington.edu/~lee/Books/riemannian.html $\endgroup$ – user254433 Jun 23 '17 at 5:10
  • $\begingroup$ @user254433 Excellent book! Jack posts here sometimes. He might have some additional suggestions. $\endgroup$ – Michael Lee Jun 23 '17 at 5:32
  • $\begingroup$ Also, if you can get your hands on a copy of Spivak's set titled A Comprehensive Introduction to Differential Geometry, that should serve you very well. $\endgroup$ – Michael Lee Jun 23 '17 at 5:37
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    $\begingroup$ I am sure everyone has their personal favourite; I would suggest picking up a few and seeing which book {\it you} get one with. For my tuppence worth, I really like Riemannian Geometry and Geometric Analysis by Jurgen Jost. $\endgroup$ – AloneAndConfused Jun 23 '17 at 7:43
  • $\begingroup$ Voyez si cette bibliographie vous est utile. On peut y ajouter les Leçons de géométrie de Postnikov. Les tomes 1 à 4 se trouvent en français, mais pas le tome sur la géométrie riemannienne, qui est en anglais ou en russe (au moins). $\endgroup$ – user49640 Jun 23 '17 at 10:46

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