Barrett O'Neill (1924– 16 June 2011) was an American mathematician. He is known for contributions to differential geometry, including two widely-used textbooks on its foundational theory. He was the author of eighteen research articles, the last of which was published in 1973. He received his Ph.D. in mathematics in 1951 from the

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...

. His doctoral advisor was

Witold Hurewicz Witold Hurewicz (June 29, 1904 – September 6, 1956) was a Polish mathematician. Early life and education Witold Hurewicz was born in Łódź, at the time one of the main Polish industrial hubs with economy focused on the textile industry. His ...

. His dissertation thesis was titled ''Some Fixed Point Theorems'' He has worked as a professor of mathematics at

UCLA The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the California ...

, where he supervised the PhDs of eight doctoral students. He made a foundational contribution to the theory of

Riemannian submersion In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces. Formal definition Let (' ...

s, showing how geometric quantities on the total space and on the base are related to one another. "O'Neill's formula" refers to the relation between the

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...

s. O'Neill's calculations simplified earlier work by other authors, and have become standard textbook material.Peter Petersen. ''Riemannian geometry.'' Third edition. Graduate Texts in Mathematics, 171. Springer, Cham, 2016. xviii+499 pp. With Richard Bishop, he applied his submersion calculations to the geometry of warped products, in addition to studying the fundamental role of convex functions and convex sets in

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...

, and for the geometry of negative

in particular. An article with his former Ph.D. student Patrick Eberlein made a number of further contributions to the Riemannian geometry of negative curvature, including the notion of the "boundary at infinity".

Major publications

Books * Barrett O'Neill. ''Elementary differential geometry.'' Revised second edition of the 1966 original. Elsevier/Academic Press, Amsterdam, 2006. xii+503 pp. * Barrett O'Neill. ''Semi-Riemannian geometry. With applications to relativity.'' Pure and Applied Mathematics, 103. Academic Press, Inc. arcourt Brace Jovanovich, Publishers New York, 1983. xiii+468 pp. * Barrett O'Neill. ''The geometry of Kerr black holes.'' A K Peters, Ltd., Wellesley, MA, 1995. xviii+381 pp. Articles * Barrett O'Neill. ''The fundamental equations of a submersion.'' Michigan Math. J. 13 (1966), 459–469. * R.L. Bishop and B. O'Neill. ''Manifolds of negative curvature.'' Trans. Amer. Math. Soc. 145 (1969), 1–49. ; * P. Eberlein and B. O'Neill. ''Visibility manifolds.'' Pacific J. Math. 46 (1973), 45–109.

References

External links

Barrett O'Neill
UCLA
Barrett O'Neill
publications on Google scholar *https://mathscinet.ams.org/mrlookup 1924 births 2011 deaths 21st-century American mathematicians Massachusetts Institute of Technology School of Science alumni 20th-century American mathematicians {{US-mathematician-stub