On integral geometry in the four dimensional complex projective space (Q1768111)

Let \(G\) be a Lie group and \(K\) a closed subgroup of \(G\). If \(M\) and \(N\) are submanifolds of the Riemannian homogeneous space \(G/K\), a central problem in integral geometry is the computation of the integral \(\int \text{vol}(M\cap gN)\,d\mu(g)\) in terms of geometric invariants of the submanifolds \(M\) and \(N\) of \(G/K\). For example, the best known classical result is the Poincaré formula for the average of the intersection number of two curves in Euclidean space. In the present article the author considers the case where \(G=U(5)\) is the five-dimensional unitary group acting on the four-dimensional complex projective space \(\mathbb{C} P^4\), \(M\) is a real four-dimensional submanifold and \(N\) is a complex two-dimensional submanifold of \(\mathbb{C} P^4\). The main result establishes the integral in terms of the volumes: \(\text{vol}(U(5))\), \(\text{vol}(N)\), \(\text{vol}(\mathbb{C} P^2)\), and another integral involving the multiple Kähler angle of \(M\) at points \(x\in M\), a concept introduced previously in an article by \textit{H. Tasaki} [Generalization of Kähler angle and integral geometry in complex projective spaces, Steps in differential geometry, Proceedings of the Colloquium on differential geometry (Debrecen 2000), 349--361 (2001; Zbl 0984.53030)].