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Riemannian geometry during the second half of the twentieth century. (English) Zbl 0928.53001
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, cannot be achieved without being familiar with quite a few examples”.
As to the contents, five main themes are dealt with in quite some detail. Firstly, the relation between the curvature of a Riemannian manifold and the topology of the underlying space is treated: topics such as pinching problems, curvature of a given sign and finiteness and compactness results fit into this framework. Secondly, space forms of different kinds and symmetric spaces are discussed in the context of a geometrical hierarchy for Riemannian manifolds. Next comes the basic question whether on a given manifold there is some ‘best’ Riemannian metric. Typically, these are metrics which are critical for certain (curvature) functionals on the space of all Riemannian metrics. Einstein metrics are one particular example and are given due attention. After a discussion of the spectrum and the eigenfunctions of the Laplacian, the author concludes with remarks on periodic geodesics and the geodesic flow. Those themes are complemented with several digressions in the main text on newly introduced concepts, and with a final section covering very briefly topics which cannot go unmentioned in a survey article of this type: so, for instance, non-compact manifolds, bundles over Riemannian manifolds, Kähler manifolds, and harmonic maps.
The author is careful to provide the reader with many references to the basic articles in every domain he touches upon, as well as to existing surveys. This is reflected in the extended bibliography of some 30 pages with up to date references. At some places though, this abundance of references and the many links to topics treated elsewhere in the text make for hard reading. Still, it pays off to make the effort as the reader is finally rewarded with an overview of the main results in a certain subfield of Riemannian geometry, as well as with a deeper insight in the complex maze of interactions between these subfields. It is thanks to Berger’s unraveling of this intricate web that the reader comes away with a deeper understanding.
As wonderful as this survey is, the reviewer would like to see a similar one dealing with the complete case and one treating local aspects of Riemannian geometry. Better even, with the author, he hopes that some ‘Handbook of differential geometry’ covering the totality of differential and metric aspects of Riemannian geometry will appear soon. While waiting, we can enjoy Berger’s paper.

53-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to differential geometry
53Cxx Global differential geometry
01A60 History of mathematics in the 20th century
53-03 History of differential geometry
58-03 History of global analysis
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C20 Global Riemannian geometry, including pinching