# Relearning differential geometry

I will shortly describe my situation and than formulate the problem. From around year I am working under supervision of my professor on master thesis in differential geometry (mainly discussion of different geometric structures on manifolds) I intend to stay in field of differential geometry during PhD. Yet I very often find myself unaware of some basic fact or have trouble to rebuild some part of theory from foundations so I have decided to relearn foundations of differential geometry. For that purpose I can spare 2-3 hours during my semester break and even longer if necessary in order to do that. I plan to workout one or two handbook covering foundations of diff. geometry (especially - here I find the biggest problems - connection theory in principal bundles) and Riemannian geometry (curvature, geodesics, normal coord., Jacobi fields etc - I know that these are basic themes yet here my gaps are quite big I presume). I have some propositions for that textbook yet I ask you for opinion or propositions if you know better one (list is partial based on: List of books I, List of books II):

1. Foundations of Differential Geometry I - S. Kobayashi, K. Nomizu (in my opinion great for connections yet lacks other material except for some Riemannian geometry)

2. Introduction to Smooth Manifolds - John Lee's (It was mentioned in second list yet however it covers a lot of material seems to basic)

3. Topics in differential geometry - P. Michor (It was not mentioned on neither of lists yest I am curious of you opinion)

My second question is about post from second list, namely it was stated there that it is worthwhile to master/ be familiar (which one??) with theory of surfaces and curves. What is your opinion on it? If it is really necessary to master that themes for knowing what is going on which book will you recommend?

I know there are many researchers here in our community and I would be very grateful for your responses. Formally I have taken two courses connected with differential geometry. Firs one rather connected with differential topology - definition of abstract manifold, construction of canonical bundles, differential forms, hodge decomposition and some facts about differential operators (I barely remember). Second one was about Riemannian geometry - Riemannian connection, geodesics, normal coordinates, Jacobi fields, curvature, Hopf-Rinow (without proof), harmonic maps (informations). Actually I have never had classes about classical diff. geometry of curves and surfaces. If it helps my current interest is in geometric structures (G-structures and pseudo-group structures) so feel free to take it into account.

• How strong is your acquaintance of multilinear algebra? – janmarqz Jul 11 '16 at 14:57
• well, I know fundamental concepts - exteror algebra, tensor algebras. Yet more from formal point of view than understanding intuition behind it. – J.E.M.S Jul 11 '16 at 18:01
• See this mo question. (I know it's too late for OP, but for those who came across this page...) – teika kazura Jul 17 '18 at 23:29