**Richard L. Bishop**

Professor Emeritus,
Department of Mathematics

University of Illinois at Urbana-Champaign

1409 W. Green Street

Urbana, Illinois 61801-2975

Office: 329 Illini Hall

Phone: office (217) 244-7339; home (217) 328-6379

FAX: (217) 333-9576

Home address: 3514 N Highcross Rd

Urbana, IL, 61802

e-mail: *Richard L. Bishop*

**General Information**

B.S. Case Institute of Technology, 1954

Ph.D. MIT, 1959

Thesis advisor: I. M. Singer

UIUC faculty member since 1959.

Visiting Appointments: UCLA, MIT

**Research**
Riemannian geometry, intrinsic metric spaces

**Ph. D. Students**

Stephanie Alexander(PhD 1967) (web page
SBA)

Larry Lipskie(PhD 1975)

Mark Thomas(PhD 1983)

Chien-Hsiung Chen(PhD 1996)

Jeffrey Ho(PhD 1999)

**Recent research paper**

In 2004 Professor Stephanie Alexander and I published a paper which gave sufficient conditions for a warped product of metric spaces to have a curvature bound, above or below. Now we have written a paper in which we establish that those sufficient conditions are also necessary. pdf

**Notable old publications**

In 1964 Richard J. Crittenden and I authored the first book which treated Riemannian geometry in a modern style which has had a lasting presence. It was published under the title "Geometry of Manifolds" in 1964 by Academic Press and reprinted in 2000 by AMS-Chelsea with some corrections. In the last chapter this book contains the original proof of an important new research result by me which is now called the Bishop Volume Theorem. Subsequently this theorem has become a key input to further research, starting with estimates on the growth of the fundamental group of a negatively curved manifold by John Milnor. It was used extensively and very effectively by M. Gromov and some authors have referred (mistakenly) to it as the Bishop-Gromov Volume Theorem. Besides being available for sale from the AMS, it also has a Google eBook version.

In Academic Year 1967-68 Barrett O'Neill and I did an extensive
project on manifolds of negative curvature. We published our work
under the title "Manifolds of Negative Curvature", 1969, Transactions
of the AMS. Up to now (2012) no electronic version has been available,
but now it can be downloaded in pdf format:

Bishop-O'Neill, 1969

Recently I have produced lecture notes in pdf form from courses I taught on Riemannian geometry and Lie groups.

The one on Riemannian Geometry uses the bases bundle and frame
bundle, as in Geometry of Manifolds, to express the geometric
structures. It has more problems and omits the background material
on differential forms and Lie groups, and the advanced material
on Riemannian imbeddings.

Riemannian geometry, July, 2013

The one on Lie groups follows the pattern of Chevalley's book
for the basics: it starts with matrix examples, then the basic
theory about the Lie group-Lie algebra relation. Then there is a
section on topological groups based on Pontryagin's treatment.
The material on representation theory ends with the Peter-Weyl theorem.
The remaining third is more unusual, covering invariants of group
actions, special functions, and actions on differential equations.

Lie groups, 2013

In 1969 I completed a research project concerning the board game
of Monopoly. Since the mathematical foundation (probability) was not
my specialty, I consulted with a late colleague, Robert B. Ash, to make
sure the terminology was correct. There were very few improvements
needed, but I asked him to be listed as a joint author. A
summary version, "Monopoly as a Markov Process", was published
in 1972 in Mathematics Magazine. The main result is a table of
expected limit frequencies for all the positions of a token.
Subsequently, this same result was obtained by computer simulation;
however, my treatment was based on a very accurate Markov process
model, from which a further analysis of the rate of convergence
could be derived. Moreover, I intended the paper to be primarily
educational, illustrating that it was feasible to calculate all the
important data for a very complicated Markov process which
had an interesting application; this is in great contrast to the
toy examples I saw in textbooks. I have no off-prints left, but a
more detailed electronic version is available here.

pdf