I'm a masters student and in about a week I begin working in a short presentation of "a theorem" by John N. Mather.
More precisely, I have to give some elements of the proof of a theorem that states something along the lines of "if the curvature of a smooth convex billiard is zero in some point, there is no invariant curve" (my rough translation from french, sorry for the mistakes, if any).
I am very vague because I know very little on the subject. My advisor teacher gives us a class on Algebraic Topology but I feel like this is more Differential Geometry.
Before the teacher gives me the documentation, I wanted to begin researching the subject but so far I'm having a hard time finding anything. If someone can point me to some articles or books in the subject I would be very thankful.
Thank you for your answers.
You can find a simple proof of this at the book:"Geometry and Billiards" by Sergei Tabachinikov, see Corollary 5.29. You can find his book at his website https://www.math.psu.edu/tabachni/Books/books.html
The proof will be a consequence of the Mirror equation, which you can also find in the same book, Theorem 5.28.
By the way this book is a great reference to learn about different aspects of convex billiards.