Riemannian geometry during the second half of the twentieth century
type: Item
instance of: scholarly article
title: Riemannian geometry during the second half of the twentieth century
zbMATH Open document ID: 0928.53001
author: Marcel Berger
published in: Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)
publication date: 1999-03-09T00:00:00Z
review text: Since its starting days back with Gauss and Riemann, the history of Riemannian geometry has been one of constant growth, both in results and in importance. Especially in the last few decades, its development has been nothing less than spectacular. As a consequence, however, there are now so many diverse directions that not even the finest expert is able to fully grasp all its different aspects. In this survey paper, Marcel Berger, one of the outstanding geometers of our time, gives his personal view on the main evolutions within the realm of Riemannian geometry. In his own words, his is ``a (biased) `elementary geometry' temper'', as well as ``a liking for results which are simple to state''. These personal preferences are reflected in the choices he makes and the restrictions he imposes: firstly, he defines Riemannian geometry quite narrowly as ``what's happening on the Riemannian manifold itself, in particular as a metric space''. Topics such as Riemannian bundles and submanifold theory are left out of the picture almost completely. Secondly, only global considerations are taken into account and the local theory is basically absent. In particular, the author restricts his attention almost exclusively to compact manifolds without boundary, even though generalizations of results to complete manifolds are quite often referred to. With these restrictions in mind, the author intends both to chronicle the history of the subject over the past fifty years and to describe the state of affairs as it is today. He avoids the all-too-easy trap of wanting to be exhaustive, but rightly focuses on those new concepts and innovative techniques that have propelled Riemannian geometry forward and that have given rise to deep results. Often, he shares his own opinion (as well as that of others) to give a most welcome point of view on the subjects treated. Let us also mention his insistence on good examples, for, as he mentions, ``to feel, to understand what a Riemannian manifold is, c…
reviewed by: Eric Boeckx
Mathematics Subject Classification ID: 53-00
Mathematics Subject Classification ID: 53Cxx
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 53-03
Mathematics Subject Classification ID: 58-03
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 58-02
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1259503
zbMATH Keywords: Riemannian geometry
zbMATH Keywords: curvature and topology
zbMATH Keywords: space forms
zbMATH Keywords: symmetric spaces
zbMATH Keywords: critical metrics
zbMATH Keywords: Laplacian
zbMATH Keywords: periodic geodesics
zbMATH Keywords: geodesic flow
zbMATH Keywords: bibliography
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1259503