Book/Slides/Videos about points-topology-manifold-charts-riemannian manifolds etc

Question

My goal is to study some kind of nonlinear systems through differential geometry. I did an intensive meeting with my supervisor in which he tried to give me an introduction and a link between the following topics:

• Points
• Topology
• Manifolds
• Charts (coordinates)
• Differential manifolds
  • Riemannian manifolds
    • Geodesics
    • Exponential maps
    • Logarithmic maps
    • $SO(d)$ case and hat-map vs v-map
  • Connections
    • Tangent bundle
    • Vector fields (flow of a vector field)
    • Christoffel's symbols
    • Levi-Civita's connection
    • Parallel transport
    • Link of the previous with the Rodrigues' formulas for rotation

Finally how to arrive from all this to Lie Brackets.

I followed during the meeting but you immediately understand that in 1 hour and a half you can't get all the details of the concepts we touched during the meeting. For this reason I would like to know if you can suggest me some books/slides/videos or whatever to get more understanding of these concepts.

I am going throught the book : Nonlinear control systems by A.Isidori.

Someone on this forum suggested to look into the work of John M.Lee: Introduction to topological Manifolds, Introduction to Smooth manifolds

I found out, reading the Isidori's book that it lacks a bit of graphical explanations which, in this context, I find really useful.

Can you help me?

Thanks a lot for the help.

Answer 1

For the first part I used: An introduction to manifolds - Loring W. Tu I like it a lot, in my opinion is a very good book with a lot of nice exercises.

For the Riemannian geometry I used the following notes

http://www.maths.tcd.ie/~dwilkins/Courses/425/RiemGeom.pdf