On the Restrictions of Rational Maps on Linear Subspaces and Some Applications in CR Geometry

We first study the rational maps between projective spaces by looking at their restrictions on linear subspaces, and then explain how the consideration of linear subspaces can be used to obtain rigidity theorems for the CR maps between real hyperquadrics. While most of the rigidity statements in this paper are well known, the proofs here are given in a manner such that the role played by linear subspaces is highlighted. We also use similar methods to prove a rigidity theorem related to the CR maps into the intersection of two real hyperquadrics of a particular simple form.