After his secondary education in Rosdorf, Gromoll entered the University of Bonn. He undertook research there with Friedrich Hirzebruch as his thesis advisor and was awarded a doctorate in 1964 after submitting his thesis Differenzierbare Strukturen und Metriken positiver krümmung auf Sphären Ⓣ. It was published in Mathematische Annalen two years later. He then spent time at Princeton in the United States and Mainz University in Germany before being appointed Miller Fellow at the University of California, Berkeley in 1966. He held this fellowship for two years, then returned to the University of Bonn for a year before being appointed to the State University of New York at Stony Brook in 1969.
During these years Gromoll published some important work. In 1968 he published Riemannsche Geometrie im Grossen Ⓣ, a set of lecture notes written jointly with W Klingenberg and Wolfgang Meyer. J W Smith writes in a review:-
These lecture notes should be of interest to a wide class of readers. The first five chapters comprise an introduction to Riemannian geometry, accessible to students with a background in real analysis, linear algebra and first concepts of general topology. ... The last two chapters include recent results of great depth. Having reached one of the most interesting frontiers of contemporary research, these notes take on the character of a high level monograph. ... The book has many excellent qualities. The authors seem to be thoroughly enjoying the mathematical ideas under consideration, and they succeed well in conveying a live appreciation for the subject. The exposition lacks neither clarity nor precision and avoids unnecessary generality. There is a nice balance between theory and application. Exercises and examples are chosen with commendable judgement, and students stand to benefit from numerous comments and observations which illuminate the subject from various directions.Also in 1968, Gromoll published a joint work with Jeff Cheeger The structure of complete manifolds of nonnegative curvature. This paper was in fact a sequel to On complete open manifolds of positive curvature which Gromoll had written with Wolfgang Meyer but did not appear in print until the following year. In the paper with Cheeger he classified the complete noncompact Riemannian manifolds of positive sectional curvature while in the sequel with Cheeger they generalised this result to the case of nonnegative curvature,
Also in 1969, the year he was appointed to Stony Brook, Gromoll had two futher papers with Wolfgang Meyer published: On differentiable functions with isolated critical points; and Periodic geodesics on compact riemannian manifolds. The second of these papers, which uses results from the first, makes an essential contribution to the problem of the existence of "many" prime closed geodesics on a compact Riemannian manifold. This is a highly significant paper but an even more important one was published by Gromoll and Cheeger in 1972, namely On the structure of complete manifolds of non-negative curvature. This paper contains a result known as the "soul theorem" which :-
... provided one of the cornerstones of the Poincaré Conjecture solution. And it actually shows up in two independent places in Perelman's proof and is a really essential ingredient in that work.In 1970, a year after his appointment at Stony Brook, Gromoll was an invited speaker at the International Congress of Mathematicians in Nice, France. His lecture Manifolds of nonnegative curvature surveyed the progress that he had made in the papers we have mentioned above. F Brickell, reviewing the published version of the lecture, writes:-
This article outlines the remarkable progress made in the study of complete but non-compact Riemannian manifolds of non-negative curvature, in the three years prior to the International Congress of 1970. Perhaps the most impressive discovery was the fact that such a manifold M is diffeomorphic to the normal bundle of a compact totally convex submanifold of M without boundary.Among many other lectures he gave to international meetings we mention the his address to the 4th Geometry Festival, UNC Chapel Hill (1988), to the 38th AMS Summer Inst., Los Angeles (1990), his Plenary Lecture at the CMS Meeting, St John (1998); and his Plenary Lecture at the 50th Anniversary of IMPA, Rio de Janeiro (2002). The Los Angeles address was entitled Spaces of nonnegative curvature. Differential geometry: Riemannian geometry and gave an overview of what was known at the time concerning manifolds of nonnegative curvature. Gerard Walschap described the lecture as:-
... a comprehensive, in-depth account not only of the progress made in this area during the past twenty years, but also of the methods that have been used.Gromoll worked at the State University of New York at Stony Brook for the rest of his career, but held a number of visiting positions over the years such as at: École Polytechnique and IHES, Paris (1975); University of Münster (1983); IMPA Rio de Janeiro (1984/1996/1997); and MSRI Berkeley (1993). Among his most significant papers written later in his career we mention (with U Abresch) On complete manifolds with nonnegative Ricci curvature (1990), (with M Dajczer) The Weierstrass representation for complete minimal real Kähler submanifolds of codimension 2 (1995), and (with G Walschap) The metric fibrations of euclidean spaces (2001). In this last mentioned paper, the authors completed the classification of metric fibrations in Euclidean space which they had begun in a paper in 1997.
Gromoll was married to Suzan: they had three children, Hans Christian, Stefan and Heidi. He died following a brain haemorrhage.
We end this biography by quoting the statement made by Gromoll when he put himself forward for the position of 'Member at large' in the American Mathematical Society in 2006:-
Having advised more than twenty Ph.D. students, I consider it a most rewarding challenge to pass on exciting new mathematics to the next generation. There are many tough issues with graduate education: We must work harder to attract more top domestic students to the field and redouble our efforts to achieve diversity. We need a better support structure for graduate students, nationally and locally. NSF is moving in only slowly, and AMS can play a bigger role beyond the current initiatives. At a different level, the serious problems with high school curricula and teacher training in Mathematics are getting more and more attention, also within AMS. As recent department chair I was involved with reshaping our local math ed program according to new state and national guidelines. I had the privilege to attend the annual AMS Education Committee meetings on several occasions, and remain very interested in these problems. Other areas of concern to me include the rapidly changing ways mathematics is disseminated and published, for instance the flood of new journals versus unlimited electronic options, and the future of our libraries.
Article by: J J O'Connor and E F Robertson