I was wondering if anybody has a suggested self-study path to understand the mathematical aspects on Hamiltonian Monte Carlo.

In this paper The Geometric Foundations of Hamiltonian Monte Carlo it is mentioned that a good reference is John Lee's Introduction to Smooth Manifolds and here is my question:

• What are the core concepts that I should know from smooth manifolds theory in order to understand Hamiltonian Monte Carlo from a mathematics perspective?

Reading Lee's book from cover to back seems a daunting task, so I'd would like to have more guidance over what sections (or topics) I should definitely read.

P.S-1: I'm not constrained to Lee's book(s), I'm just looking for a kind of "syllabus" of the core topics to understand HMC.

P.S-2:If it is helpful, I have background in measure-theoretic probability, real analysis and introductory topology.

Thanks for your answers!