Valuations and Curvature Measures on Complex Spaces

Abstract

We survey recent results in hermitian integral geometry, i.e. integral geometry on complex vector spaces and complex space forms. We study valuations and curvature measures on complex space forms and describe how the global and local kinematic formulas on such spaces were recently obtained. While the local and global kinematic formulas in the Euclidean case are formally identical, the local formulas in the hermitian case contain strictly more information than the global ones. Even if one is only interested in the flat hermitian case, i.e. C n , it is necessary to study the family of all complex space forms, indexed by the holomorphic curvature 4λ , and the dependence of the formulas on the parameter λ . We will also describe Wannerer's recent proof of local additive kinematic formulas for unitarily invariant area measures.

Kinematic formulae for tensorial curvature measures

“…; see ( 8) and (11) in the proof of Theorem 1. It should also be observed that the functionals φ r,s−2m,l+i n−1 can be expressed in terms of the functionals…”

“…Moreover, in the case k = n we use that φ r,s−2m,l+i n vanishes for m = s 2 . Hence, for even s we have to simplify the sum i Γ(i + l − 1) Γ( n 2 + l + i) × φ r,0, s 2 +l n (P, β)φ j (P ′ , β ′ ) = c s n,j φ r,0, s 2 +l n (P, β)φ j (P ′ , β ′ ), (11) as required. This completes the proof.…”

“…Various other groups of isometries, also in Riemannian isotropic spaces, have been studied in recent years. Major progress has been made, for instance, in Hermitian integral geometry (in curved spaces), where the interplay between global and local results turned out to be crucial (see [13,14,25,26,79,80,76] and the survey [11]), but various other group actions have been studied successfully as well (see [4,8,9,12,17,18,19,23]).…”