Integral geometry on submanifolds of dimension one and codimension one in the product of spheres (Q1431083)

Let \(M\) and \(N\) be submanifolds of \(S^{m+1} \times S^{n+1}\) of dimension one and codimension one, respectively, and assume that for almost all \(g \in G = \text{SO}(m+2) \times \text{SO}(n+2)\) the submanifolds \(M\) and \(gN\) intersect transversely. The author derives an explicit description of the integral \(\int_G \#(M \cap gN) \,d\mu_G(g)\) in terms of the volumes of \(G\), \(S^{m+1}\) and \(S^{n+1}\) and of the Gauss hypergeometric function.