On the spectrum and the geometry of spherical space forms. I
type: Item
instance of: scholarly article
title: On the spectrum and the geometry of spherical space forms. I
zbMATH Open document ID: 0547.58040
DOI: 10.1007/BF01000336
author: Friedbert Prüfer
published in: Annals of Global Analysis and Geometry
publication date: 1985-01-01T00:00:00Z
review text: In this paper connected compact manifolds M of constant sectional curvature one are studied. Using the method of the Euler-Poisson-Darboux equation an explicit Poisson formula for the distribution \(\zeta_ n:=\sum\cos \sqrt{\mu +(n-1)^ 2/4}\cdot, n:=\dim M\) was derived. Here the sum runs through all \(\mu \in spec(M)\). This formula describes the relation between \(\zeta_ n\) and geometrical data of M. Furthermore the lengths of closed geodesics belonging to a free homotopy class are calculated. At the end of the paper a short proof of the statement \(\sin g \sup p \zeta_ n=\{0\}\cup L(M)\) is given, where L(M) is the set of all periods of the periodic geodesics c of M normed by \(\|\dot c\| =1\).
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 3872349
zbMATH Keywords: Euler-Poisson-Darboux equation
zbMATH Keywords: Poisson formula
zbMATH Keywords: closed geodesics
zbMATH Keywords: periodic geodesics
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01000336
OpenAlex ID: W4249414195
description: scientific article; zbMATH DE number 3872349