Richard Lawrence Bishop (August 12, 1931 – December 18, 2019) was an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

who specialized in

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

and taught at the

University of Illinois at Urbana–Champaign The University of Illinois Urbana-Champaign (UIUC, U of I, Illinois, or University of Illinois) is a public land-grant research university in the Champaign–Urbana metropolitan area, Illinois, United States. Established in 1867, it is the f ...

. Bishop went to

Case Institute of Technology Case Western Reserve University (CWRU) is a Private university, private research university in Cleveland, Ohio, United States. It was established in 1967 by a merger between Western Reserve University and the Case Institute of Technology. Case ...

as an undergraduate, earning a B.S. in 1954. Next he earned his Ph.D. from the

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...

in 1959, and immediately joined the UIUC faculty, where he stayed until his retirement in 1997. His thesis, ''On Imbeddings and Holonomy'', was supervised by

Isadore Singer Isadore Manuel Singer (May 3, 1924 – February 11, 2021) was an American mathematician. He was an Emeritus Institute Professor in the Department of Mathematics at the Massachusetts Institute of Technology and a Professor Emeritus of Mathemat ...

. At UIUC, his doctoral students included future UIUC colleague

Stephanie B. Alexander Stephanie Brewster Brewer Taylor Alexander (September 1, 1941 – November 20, 2023) was an American mathematician, a professor of mathematics at the University of Illinois at Urbana–Champaign. Her research concerned differential geometry and me ...

. He is the author of ''Geometry of Manifolds'' (with Richard J. Crittenden, AMS Chelsea Publishing, 1964, translated into Russian 1967 and reprinted 2001) and ''Tensor Analysis on Manifolds'' (with Samuel I. Goldberg, Macmillan, 1968, reprinted by Dover Books on Mathematics, 1980). In 2013, Bishop became one of the inaugural

fellow A fellow is a title and form of address for distinguished, learned, or skilled individuals in academia, medicine, research, and industry. The exact meaning of the term differs in each field. In learned society, learned or professional society, p ...

s of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

.List of Fellows of the American Mathematical Society
retrieved 2014-06-16. The

Bishop–Gromov inequality In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness ...

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

, one form of which appeared in his book with Crittenden, is named after him and Mikhail Gromov, who gave an improved formulation of Bishop's result. He introduced the "Bishop frame" of curves in Euclidean space, an alternative to the better-known Frenet frame. With

Barrett O'Neill 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 ...

he made foundational contributions to the study of convex functions and convex sets in Riemannian geometry and their applications in the study of negative

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 po ...

, including to the geometry of warped products.

Notable publications

* R.L. Bishop and B. O'Neill. ''Manifolds of negative curvature.'' Trans. Amer. Math. Soc. 145 (1969), 1–49. ; * Richard L. Bishop
''There is more than one way to frame a curve.''
Amer. Math. Monthly 82 (1975), 246–251. * Richard L. Bishop and Richard J. Crittenden. ''Geometry of manifolds.'' Reprint of the 1964 original. AMS Chelsea Publishing, Providence, RI, 2001. xii+273 pp. ;

References

1931 births 2019 deaths 20th-century American mathematicians 21st-century American mathematicians Case Western Reserve University alumni Massachusetts Institute of Technology alumni University of Illinois Urbana-Champaign faculty Fellows of the American Mathematical Society People from Allegan, Michigan {{US-mathematician-stub