Herbert Busemann. Selected Works, vol. I
Résumé
The present volume contains Busemann’s papers on the foundations of geodesic
metric spaces and the metric geometry of Finsler spaces. The papers
are preceded by a short biography, six essays on various aspects of Busemann’s
work, and reviews of two of his books, namely, Metric Methods in Finsler
Spaces and in the Foundations of Geometry and The geometry of Geodesics.
The first essay is meant to help the reader getting an idea of the content
of the three very early papers written by Busemann on metric geometry and
the foundations, and it may act a reading guide for these three articles. The
articles were written in German and they remain practically unknown. An
English translation of them is provided in the present edition. The first of
these articles consists in Busemann’s doctoral dissertation. Its main theme
is the foundations of the metric theory of Minkowski spaces, a theme that
accompanied Busemann for the rest of his life. The two other papers are
elaborations and complements on the same theme. In these three papers,
we already find some of the essential ideas of Busemann on metric geometry,
in particular the introduction of the so-called Busemann functions, the
global consequences of the differentiablilty and the convexity properties of
spheres and horospheres in geodesic metric spaces, and the role played by
the analogue of Desargues’ theorem in the axioms of geometry. In their 1993
paper Novel results in the geometry of geodesics, Busemann and Phadke recall
that “G-spaces were first introduced without this name by Busemann
in his thesis in 1931.”
The second essay is concerned with Desargues’ theorem in Busemann’s
work.
The third and fourth essays are concerned with the open problems that
Busemann formulated in metric geometry, in particular his conjectures on
the finite-dimensionality and the homogeneity of G-spaces, and the conjecture
stating that every n-dimensional G-space is a topological manifold.
Partial solutions were given to these problems under various conditions of
boundedness of metric curvature. Several of Busemann’s problems are mentioned
in his Geometry of geodesics. Some of them have been solved, others
have been solved in part, and others remain intact. The two essays on the
problems that are contained in the present volume discuss the history and
the literature around these problems and provide updated lists of open problems
on this theory. These two essays are also the occasion of comparing
Busemann’s approach to metric geometry, and in particular his notion of
curvature, to the theory that was developed during the same period in the
Soviet Union by A. D. Alexandrov and the school he founded there.
The fact that some of Busemann’s problems were solved under additional hypotheses
on the boundedness of the metric curvature, on the one hand in the sense
of Alexandrov and on the other hand in the sense of Busemann, are good
examples of the marriage between the two theories.
The two remaining essays concern a subject on which Busemann worked
and which is very poorly known, namely, his theory of timelike spaces. This
is a purely metric theory which is analogous to his theory of G-spaces, but
instead of having Riemannian and Finsler spaces in the background, the
theory is a generalization of semi-Riemannian geometry, that is, manifolds
carrying, on their tangent space, a field of quadratic forms of signature
(−,+, . . . , +). From the physical point of view, this theory provides a metrical
setting for the theory of relativity. The first of the two introductory
essays is a survey of Busemann’s work on this subject, and the second one
is a survey on chronogeometry, a subject which is very close to Busemann’s
timelike space geometry, which was developed by Alexandrov and his
school. The comparison between the two theories shows the differences
and similarities between the approaches. As a matter of fact, Busemann
wrote two papers on timelike spaces and he had two students working on
this subject. In contrast, Alexandrov had several dozens of students and
collaborators working on chronogeometry over a period of more than three
decades, with regular seminars and a large number of publications on the
subject. This state of affairs is not limited to timelike spaces and chronogeometry,
it concerns Busemann’s school in general, compared to that of
Alexandrov.
In this volume, Busemann’s papers are given in chronological order. Some
of them, like On the foundations of calculus of variations (1941) written in
collaboration with W. Mayer, Local metric geometry (1944) and Spaces with
non–positive curvature (1948), are foundational papers, and present the theory
in detail. The collection of papers also includes two surveys, namely,
The geometry of Finsler spaces (1950), The synthetic approach to Finsler
spaces in the large (1961), a paper that Busemann wrote at the occasion of a
CIME course, and Novel results in the geometry of geodesics (1993) written
in collaboration with Busemann’s former student B. B. Phadke. The CIME
paper was written well after the two foundational books by Busemann, Metric
methods in Finsler spaces and in the foundations of geometry (1942) and
The geometry of geodesics (1955), and the paper with Phadke was written
30 years after Busemann’s first paper on the subject. It constitutes the last
exposition by Busemann of the state of the art in the theory he founded.
Reading these surveys will give the reader a good idea of the development
of the theory.
Skimming through all these papers of Busemann will give the reader a
feeling of the coherence and unity of his work. Going more thoroughly into
these papers, the reader will discover a source of great ideas, at some places
highlighted by Busemann, but at some other places hidden in remarks and
examples.