On a relation between the total curvature and the measure of rays
type: Item
instance of: scholarly article
title: On a relation between the total curvature and the measure of rays
zbMATH Open document ID: 0531.53035
author: Koichi Shiga
published in: Tsukuba Journal of Mathematics
publication date: 1982-01-01T00:00:00Z
review text: A well-known theorem of Cohn-Vossen says that for a non-compact complete surface M the following inequality holds \(\int K\leq 2\pi e(M)\), provided the total curvature \(\int K\) exists. Here K is the Gaussian curvature and e(M) is the Euler characteristic of M. In this paper the author proves that 2\(\pi\) e(M)-\(\int K\) is an upper bound for the measure of all initial unit vectors of all the geodesic rays (length minimising) emanating from a given point. He also proves that if M is homeomorphic to \({\mathbb{R}}^ 2\), then 2\(\pi\)-\(\int K_+\) is a lower bound for this measure, where \(K_+=\max(K,0)\). The method of proof is application of the Gauss-Bonnet formula to a well chosen sequence of compact domains exhausting M. The proof relies on classical ideas of Cohn-Vossen and Busemann.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 3842664
zbMATH Keywords: total curvature
zbMATH Keywords: Gauss-Bonnet formula
zbMATH Keywords: Euler characteristic
zbMATH Keywords: geodesic rays
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496159448
OpenAlex ID: W2291989195
DOI: 10.21099/TKBJM/1496159448
description: scientific article; zbMATH DE number 3842664