The canonical geodesic involution and harmonic spaces
type: Item
instance of: scholarly article
title: The canonical geodesic involution and harmonic spaces
zbMATH Open document ID: 0532.53015
DOI: 10.1007/BF02329741
published in: Annals of Global Analysis and Geometry
publication date: 1983-01-01T00:00:00Z
review text: Let (M,g) be an analytic Riemannian manifold. If one calculates the power-series expansion for the mean curvature of a geodesic sphere (with respect to the radius), it can be shown that the coefficients are not independent functions on M. In fact, each odd coefficient can be calculated from the previous coefficients and their derivatives. It is the aim of this paper to give an explicit recursion formula. One gets hence an alternative proof of the author's previous result stating that the ''odd harmonicity conditions'' are consequences of the ''even harmonicity conditions'' [see the book by \textit{H. S. Ruse}, \textit{A. G. Walker} and \textit{T. J. Willmore}, Harmonic spaces (1961; Zbl 0134.392)]. As a by-product, the author shows that the characteristic polynomial of the metric tensor calculated in normal coordinates is invariant with respect to the canonical involution. So far, this was known only for the last coefficient of this polynomial, i.e., for the volume density function [see Chapter VI in: \textit{A. Besse,} Manifolds all of whose geodesics are closed (1978; Zbl 0387.53010)].
Mathematics Subject Classification ID: 53B20
zbMATH DE Number: 3844544
zbMATH Keywords: harmonic spaces
zbMATH Keywords: power-series expansion
zbMATH Keywords: mean curvature
zbMATH Keywords: geodesic sphere
author: Lieven Vanhecke
reviewed by: Oldřich Kowalski
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02329741
OpenAlex ID: W1996842962
description: scientific article; zbMATH DE number 3844544