A new expression for the density of totally geodesic submanifolds in space forms, with stereological applications
A new expression of the density \(dL^n_r\) of the \(r\)-planes \(L^n_r\) in the \(n\)-dimensional simply connected space form \(M^n_\lambda\) with constant sectional curvature \(\lambda\) is shown, namely the formula \(dL^n_r=c_\lambda(\rho)^r s_\lambda(\rho)^{n-2r-1}d\sigma_{r+1}dL^n_{r+1[O]}\) with \[ s_\lambda(\rho)= \begin{cases} \lambda^{-\frac 12}\sin(\rho\sqrt\lambda), & \lambda>0,\\ \rho, & \lambda =0,\\ |\lambda|^{-\frac 12}\sinh(\rho\sqrt{|\lambda|}), & \lambda<0,\end{cases} \quad \text{and}\quad c_\lambda(\rho)=\begin{cases} \cos(\rho\sqrt\lambda), & \lambda>0,\\ 1, & \lambda=0,\\ \cosh(\rho\sqrt{|\lambda|}), & \lambda<0.\end{cases} \] Hereby \(d \sigma_{r+1}\) is the volume element in the \(r+1\)-plane \(L^n_{r+1[O]}\) through a fixed point \(O\) and \(L^n_r\) corresponds to the nearest point of \(L_r\) from \(O\). The authors apply this decomposition to rewrite Santalo's sectional formula, and thereby they obtain new mean values for lines meeting a convex body.
scientific article; zbMATH DE number 5509730