Minkowski Space On The Quadratic Surface Complete Poncelet Theorem

Poncelet's theorem is a famous theorem in geometry which refers to a straight line reflects elastically on an ellipse and the properties it possesses. Especially, if the trajectory of the line forms a closed polygon after p bounces on the boundary, then we can obtain Poncelet's theorem in two-dimensional space, and the polygon is called Poncelet's polygon.We can relate this theorem to some practical physical problems: one particle moves freely in some space and bounces elastically on a quadric boundary in this space, then we can treat the free particle as the straight line and the reflection as the bounce on boundary. These systems are billiard systems. The similarity between Poncelet's theorem and quadric billiard system facilitate us using these results to describe the properties of the particle. We can also extend Poncelet's theorem to high-dimensional spaces and derive corresponding results on restricted quadric surfaces.By the development of superstring theory and cosmology, dynamical properties of AdS spaces and dS spaces attract more and more interest. Because of AdS spaces and dS spaces can be treat as quadric surfaces in high-dimensional Minkowski spaces, we can analyse the motion of a free particle in these spaces by Poncelet's theorem.The main results of this thesis are as follows:From different perspectives, we discuss the reflection of a free particle on a hyperplane in a Minkowski space, and derive the corresponding reflection law which is consistent with its definition in projective geometry. Then we generalise Poncelet's theorem and full Poncelet's theorem to high-dimensional Minkowski spaces and their quadric surfaces including AdS spaces and dS spaces.In addition, I also introduce some of my work on quantum information in appendix.