[Math] Reference request for some topics in Differential Geometry like connections, metrics, curvature etc.

Question

I wish to study some topics in Differential Geometry like Vector bundles, Metrics, Connections, Curvature,Geodesics, Holonomy, Characteristic classes etc. I am considering reading this book ' Differential Geometry' by Clifford Henry Taubes but I am not sure whether it is a good book. Please advise. What are other good textbooks to learn these topics from ? I have studied point set topology and basic manifold theory ( tangent spaces,vector fields, differential forms,integration, and basic de Rham Theory) but have no knowledge of Algebraic Topology. I am aware of the books 'Fibre Bundles' by Dale Husemoller and 'Characteristic Classes' by Milnor but these might require Algebraic Topology as a prerequisite.

Answer 1

Lee's books, as mentioned in previous posts are excellent indeed. For an introduction to Riemannian Geometry I warmly recommend Riemannian Geometry by Do Carmo. For "topics in Differential Geometry" as you put it, there is a marvelous book by Peter Michor called just that, Topics in Differential Geometry. It's an excellent book that covers just what you are looking for.

Furthermore, there are two very nice books by Morita called Geometry of Differential Forms and Geometry of Characteristic Classes.