In this semester I want to study Conformal Differential Geometry, but I don't know which book is suitable as a textbook or reference.
Can people recommend textbooks and/or other references to me? If there are some online videos for the lecture, it would be more friendly to me.
I would recommend starting with an inspirational lecture of Charles Fefferman on Conformal Invariants on Youtube.
Next, one can brush up the necessary basics by reading wonderful lectures notes of Jeff A. Viaclovsky Math 865, Topics in Riemannian Geometry.
To get the taste of the modern conformal differential geometry, you should try to read Michael Eastwood's Notes on conformal differential geometry. This is not easy reading, though. You will have a lot of questions, that you may need to ask either here or on even on MathOverflow.
To make more sense of Eastwood's notes, it would be helpful to give a good read to An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity by Sean Curry and A. Rod Gover.
From this video recording of A. Rod Gover' talk Boundary operators on conformally compact manifolds, and a boundary Loewner-Niremberg-Yamabe problem you can get an idea of what the latest results in conformal differential geometry are about.
This list is biased, incomplete, opinionated, based on my personal experience, and may be misleading and outdated, so please beware of the risks.
