Transferred kinematic formulae in two point homogeneous spaces (Q2479949)

The Poincaré formula of integral geometry expresses the integral of the number \(n\) of intersection points of a moving curve \(c_1\) and a fixed curve \(c_2\) of the Euclidean plane with respect to the kinematic measure \(\text{ d}K\) over all possible positions of \(c_1\) in terms of the curve lengths \(L_1\) and \(L_2\): \(\int n \,\text{ d}K = 4L_1L_2\). More generally, if \(G/K\) is a two point homogeneous space of dimension \(n\) with submanifolds \(M\) and \(N\) of respective dimensions \(p\) and \(n-1\) and with finite volume, the formula \[ \int_G \text{vol}(M \cap gN) \;\text{ d}\mu_G(g) = \frac{\text{vol}(K)\;\text{vol}(S^{p-1})\;\text{vol}(S^n)} {\text{vol}(S^p)\;\text{vol}(S^{n-1})} \text{vol}(M) \text{vol}(N), \] a generalization of the Poincaré formula, due to \textit{R. Howard} [Mem. Am. Math. Soc. 509 (1993; Zbl 0810.53057)] holds. Here, the author derives a similar formula for hypersurfaces \(M\), \(N\) and homogeneous integral invariants \(I^{{\mathcal W}_2}\), \(I^{{\mathcal U}_{n-2}}\) of degree 2: \[ \int_G I^{{\mathcal W}_2} (M \cap gN) \; \text{ d}\mu_G(g) = V \cdot a(n-1,n-1,n)(I^{{\mathcal W}_2}(M)\text{vol}(N) + \text{vol}(M)I^{{\mathcal W}_2}(N)), \] \[ \int_G I^{{\mathcal U}_{n-2}} (M \cap gN) \; \text{ d}\mu_G(g) = V \cdot b(n-1,n-1,n)(I^{{\mathcal U}_{n-2}}(M)\text{vol}(N) + \text{vol}(M)I^{{\mathcal U}_{n-2}}(N)), \] where \(V = \frac{\text{vol}(K)}{\text{vol}(SO(n))}\). The constants \(a(p,q,n)\) and \(b(p,q,n)\) were determined by \textit{H. J. Kang, T. Sakai}, and \textit{Y. J. Suh} [Indiana Univ. Math. J. 54, 1499--1519 (2005; Zbl 1093.53079)]. In the course of the proof the author develops and uses a generalization of the transfer principle of integral geometry that allows the transfer of kinematic formulas from one homogeneous space to any other with the same isotropy subgroup.