Integral geometry on product of spheres (Q1860630)

Let \(G/K\) be a Riemannian homogeneous space with a \(G\)-invariant Riemannian metric, and let \(M\) and \(N\) be submanifolds of \(G/K\) with \(\dim M+\dim N=\dim G/K\). Assume that \(G\) is unimodular, and for almost all \(g\in G\), \(M\) and \(gN\) intersect transversely. In the paper under review the author focuses on the Poincaré-type formula obtained by \textit{R. Howard} [Mem. Am. Math. Soc. 509 (1993; Zbl 0810.53057)], in the interesting case \(G/K=S^{2}\times S^{2}\), \(\dim M=1\) and \(\dim N=3\). For special choices of \(M\) and \(N\), further results are given.