On integral geometry in the four dimensional complex projective space
type: Item
instance of: scholarly article
title: On integral geometry in the four dimensional complex projective space
zbMATH Open document ID: 1085.53067
author: Hong Jae Kang
published in: Tokyo Journal of Mathematics
publication date: 2005-03-14T00:00:00Z
review text: 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)].
reviewed by: Salvador D. Gigena
Mathematics Subject Classification ID: 53C65
zbMATH DE Number: 2145299
zbMATH Keywords: generalized Kähler angle
zbMATH Keywords: complex submanifold
zbMATH Keywords: Poincaré formula
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208397
OpenAlex ID: W2073193642
cites work: Slant immersions
cites work: Slant submanifolds in complex Euclidean spaces
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cites work: On conjugate loci and cut loci of compact symmetric spaces. I
cites work: Generalization of Kähler angle and integral geometry in complex projective spaces
cites work: Q4788893
DOI: 10.3836/TJM/1244208397
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description: scientific article; zbMATH DE number 2145299