Angle between two complex lines

Abstract

The chordal distance function on a complex projective space algebraically defines an angle between any two complex lines, which is known as the Hermitian angle. In this expository paper, we show that one can canonically construct a real line corresponding to each of these complex lines so that the real angle between these two real lines exactly agrees with the Hermitian angle between the complex lines. This way, the Hermitian angle is interpreted as a real angle, and some well known results pertaining Hermitian angles are proved using real geometry. As an example, we give a direct and elementary proof that the chordal distance function satisfies the triangle inequality.