In the previous academic term I had the pleasure to teach Analysis on Manifolds, essentially an introduction to differential geometry for third year students.

In the course I tried to address the topic with the perspective that some of the students would be joining also my master course in Hamiltonian Mechanics later on next year. So many of the topics go in the direction of introducing the prerequisites for studying flows, symplectic geometry and sub-Riemannian geometry.

From the feedback I got, it seems that the course turned out nice – even thought there will be a number of changes needed to improve it next year – so I decided to remove the unfinished parts from the lecture notes and publish them here for wider availability under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

You can follow its updates by monitoring the corresponding Releases page. Once I believe that they are self-consistent and polished enough, I surely plan to rewrite and simplify the whole orientation chapter and add more on metrics and connections, I will publish it on the AMS open math notes like I did with the notes from Hamiltonian Mechanics.

As usual PR, feedback and corrections are always welcome. Please make sure to acknowledge your contribution in the introduction if you contribute.