We will follow the book "An Introduction to Manifolds" by Loring W. Tu. (Second Edition).
§1-9, 11.1-11.4, 12.1, 12.2, §13-16, 17.1-17.6, 18.1-18.7, §19, 21.1-21.4, §22, 23.1-23.6, §24, 25.1-25.3, 26.1-26.23, 27.1-27.4, §29, Appendices B.3, §C.
From Loring W. Tu, "Differential Geometry": Sections 15.4 - 15.7 (six pages). I recommend downloading this book from the library website and printing the few pages we need.
Through the university's web system, you can access the book here:
https://link.springer.com/book/10.1007%2F978-1-4419-7400-6