Integral Geometry in Hermitian Spaces

Abstract

where Skl O if k,7c/z and 1 if k7 = . Since the coefficients e;k are complex numbers, the group U depends upon (n + 1)2 real parameters. The geometry of Pn with the fundamental group U of transformations is called the Hermitian geometry (more precisely the elliptic hermitian geometry) and the space Pn itself is called an Hermitian space. The Integral Geometry in these spaces was initiated by Blaschke [2] who gave the densities for linear subspaces, without making applications to integral formulas. The case n = 2 was first considered by Varga [8] and later, in a more complete form, by Rohde [6]. In the present paper we generalize to the n-dimensional case some of the results which Varga and Rohde obtained for the plane. The main results we obtain are the following: We first determine the explicit form of the left invariant element of volume du of the group U. If Lr? is a fixed linear subspace of dimension r (throughout the paper we shall mean by dimension the complex dimension)