# In memoriam: Barrett O'Neill Professor of Mathematics, Emeritus, 1924 – 2011

Professor Emeritus Barrett O'Neill died on June 16, 2011, at age 87. O'Neill joined the department in 1951, directly from MIT, where he had just received his PhD under the direction of Witold Hurewicz. O'Neill retired in 1991, but he continued his mathematical work, with a major book on relativity, The Geometry of Kerr Black Holes, published in 1995. O'Neill began his mathematical life as an algebraic topologist: his dissertation was on fixed point theory and he made further contributions to that subject, developing a generalization of the Lefschetz Fixed Point Theorem to multi-valued (set-valued) mappings. But quite early on, he turned primarily to Riemannian geometry and to semi-Riemannian geometry, the geometry of non-degenerate quadratic forms on the tangent spaces that are not positive definite.

O'Neill had a long and distinguished career, exerting a notable influence on his fields of work. The list of citations of his work is enormous, with over 1,000 citations in the Science Citation Index.

In a famous paper on Riemannian submersions, he developed a formula, now known as ONeill's formula, showing that the sectional curvature of the base manifold was always at least as large as the associated vertically-lifted sectional curvature of the larger-dimensional domain manifold. The elegance and power of this paper brought the idea of Riemannian submersion into a prominent position, which it has retained ever since, and is the fundamental tool in particular in the construction of examples of manifolds of positive curvature.

A second highly influential work was O'Neill's investigation of manifolds of negative curvature. His paper with R.L. Bishop on this topic introduced a new viewpoint on the whole subject and also formalized the idea of warped products and bundles, an idea that has been of vital importance also in the construction of manifolds of positive curvature. In a second paper on negative curvature, joint work with his student, P. Eberlein, O'Neill introduced the important concepts of visibility and of geometry at infinity for negatively curved manifolds. These two papers together revitalized the subject of manifolds of negative curvature and led to a long sequence of developments by Eberlein, Ballmann, Spazier, Gromov, among others.

O'Neill wrote three books, each of great distinction. One was an undergraduate text book on differential geometry, Elementary Differential Geometry, which was notable for its systematic use of differential forms - unusual in an undergraduate book at that time - and also notable for the elegance of its illustrations at a time before computer graphics. His bookSemi-Riemannian Geometry was one of the first books to treat indefinite metric geometry systematically on an equal footing and in modern notation with the more usual positive definite Riemannian geometry. This book was so superbly done that it has often been used as a text even when only the Riemannian case is at issue. His last book The Geometry of Kerr Black Holes is a masterpiece of exposition and mathematical insight, managing to get to deep matters indeed while being accessible to readers with only limited background in differential geometry, so that its readership has been very broad. O'Neill used to remark that whereas he had written his first two books for the benefit of other people, he had written The Geometry of Kerr Black Holes to please himself. Fortunately, he ended up pleasing a wide variety of readers as well.

O'Neill had eight PhD students, several of whom became distinguished geometers (Patrick Eberlein, Alfred Gray).

O'Neill was an encouraging, intellectually stimulating, and cheerful presence in our department, who was in particular extraordinarily kind and helpful to his younger colleagues. He had a fine dry sense of humor. He delighted, for example, in a wry way in showing people a copy of a book in which another author had unceremoniously stolen without credit the remarkable illustrations in his own Elementary Differential Geometry. The slings and arrows of life never dimmed his enthusiasm for it and for the life of mathematics in particular. He will be much missed by us all.

He is survived by his wife Hope, their three children Eric, Evelyn, and Jean, and two grandchildren.

Robert E. Greene and Peter Petersen **A memorial tribute to O'Neill will be held on Thursday, September 15, 2011, 2:00-4:00pm, at the UCLA Faculty Center. A light reception will follow.**

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