A new expression for the density of totally geodesic submanifolds in space forms, with stereological applications
type: Item
instance of: scholarly article
title: A new expression for the density of totally geodesic submanifolds in space forms, with stereological applications
zbMATH Open document ID: 1168.53039
author: J. Martínez
published in: Differential Geometry and its Applications
publication date: 2009-02-20T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 52A55
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 5509730
zbMATH Keywords: convex body
zbMATH Keywords: integral geometry
zbMATH Keywords: Santaló's sectioning formula
zbMATH Keywords: space form
zbMATH Keywords: stereology
zbMATH Keywords: support set
zbMATH Keywords: submanifold
reviewed by: Kurt Leichtweiß
MaRDI profile type: Publication
full work available at URL: j.difgeo.2008.06.013
OpenAlex ID: W2028848459
Wikidata QID: Q115357457
cites work: Q4024093
cites work: Integral geometry and geometric inequalities in hyperbolic space
cites work: Local Stereology
cites work: Q3291025
cites work: Measuring the surface area of a convex body
cites work: Q4110292
cites work: Integralgeometrie 3. Croftons Formeln für den Raum
DOI: 10.1016/J.DIFGEO.2008.06.013
description: scientific article; zbMATH DE number 5509730