Richard Lawrence Bishop (1931–2019) was a distinguished American mathematician whose work in differential geometry helped shape the modern landscape of the field. A long-time professor at the University of Illinois at Urbana-Champaign, Bishop is best known for his fundamental contributions to Riemannian geometry—most notably the volume comparison theorem that bears his name—and for authoring textbooks that became the standard for generations of graduate students.
1. Biography: Early Life and Academic Career
Richard L. Bishop was born on August 20, 1931, in Jerome, Michigan. He demonstrated an early aptitude for the physical sciences and mathematics, eventually enrolling at the Case Institute of Technology (now Case Western Reserve University), where he earned his B.S. in 1954.
He moved to the Massachusetts Institute of Technology (MIT) for his doctoral studies during a "golden age" of geometry. Under the supervision of the legendary Isadore M. Singer (co-founder of the Atiyah-Singer Index Theorem), Bishop completed his Ph.D. in 1959. His dissertation focused on the geometry of manifolds, a theme that would define his professional life.
In 1959, Bishop joined the faculty of the University of Illinois at Urbana-Champaign (UIUC). He remained at UIUC for the duration of his career, rising to the rank of full professor and eventually becoming Professor Emeritus upon his retirement in 1997. He passed away on December 18, 2019, in Urbana, Illinois, leaving behind a legacy as a meticulous researcher and a dedicated mentor.
2. Major Contributions
Bishop’s work focused on the relationship between the curvature of a space and its global topological and geometric properties.
The Bishop Volume Comparison Theorem
This is his most celebrated achievement. Proved in 1963, the theorem provides a way to compare the volume of a "ball" in a curved Riemannian manifold to the volume of a ball in a space of constant curvature (like a flat plane or a perfect sphere). It essentially shows that if a space is "more curved" than a reference space, its volumes grow more slowly. This theorem became a cornerstone of modern geometric analysis.
Warped Products (Bishop-O’Neill Theory)
In a 1969 collaboration with Barrett O’Neill, Bishop introduced the concept of "warped products." This mathematical framework allows researchers to construct new manifolds by "warping" the metric of a product space. This has had a profound impact on General Relativity, as many exact solutions to Einstein’s field equations (including Schwarzschild and Robertson-Walker metrics) are described using warped products.
Manifolds of Non-Positive Curvature
Bishop conducted pioneering research into the structure of manifolds where the curvature is zero or negative. His work helped establish how the "local" property of being curved affects the "global" shape and connectivity of the entire space.
The Bishop Frame (Parallel Transport Frame)
In the study of curves in three-dimensional space, Bishop proposed an alternative to the standard Frenet-Serret frame. The "Bishop Frame" is useful because it remains well-defined even when the curvature of a curve vanishes (which causes the Frenet frame to fail), making it a vital tool in computer graphics and DNA modeling.
3. Notable Publications
Bishop was a prolific author whose books are praised for their clarity and rigor.
- "A relation between volume and curvature" (1963): Published in the Proceedings of the American Mathematical Society, this paper introduced the foundational version of the Bishop Volume Comparison Theorem.
- "Geometry of Manifolds" (1964): Co-authored with Richard J. Crittenden. For decades, this was the definitive graduate-level text on differential geometry, bridging the gap between classical surface theory and modern global analysis.
- "Tensor Analysis on Manifolds" (1968): Co-authored with Samuel I. Goldberg. This book remains in print (via Dover Publications) and is highly regarded for its accessibility to both mathematicians and physicists.
- "Manifolds of negative curvature" (1964): Published in the Transactions of the American Mathematical Society, this work explored the topological constraints imposed by negative curvature.
4. Awards and Recognition
While Bishop was a humble figure who avoided the spotlight, his peers recognized the immense utility of his work:
- Sloan Research Fellowship (1966–1968): Awarded to early-career scientists showing outstanding promise.
- The Bishop-Gromov Inequality: In the 1980s, Mikhail Gromov generalized Bishop’s work to a global setting. The fact that the result is universally known as the "Bishop-Gromov inequality" cements Bishop’s status as a primary architect of the field.
- Emeritus Status: Granted by UIUC in 1997 in recognition of nearly 40 years of service to the university and the mathematical community.
5. Impact and Legacy
Richard Bishop’s influence is felt most strongly in the fields of Riemannian Geometry and Geometric Analysis. His volume comparison theorem is not just a result but a tool used daily by researchers to prove the existence of solutions to partial differential equations on manifolds.
Furthermore, his work on warped products provided the mathematical language necessary for physicists to describe the expansion of the universe and the geometry of black holes. In the digital age, his "Bishop Frame" has found a second life in Computer-Aided Geometric Design (CAGD) and robotics, where it is used to calculate smooth paths for camera movements and robotic arms.
6. Collaborations
Bishop’s most significant research was often the result of fruitful partnerships:
- Samuel I. Goldberg: A colleague at UIUC with whom he wrote the influential Tensor Analysis on Manifolds.
- Richard J. Crittenden: His co-author on Geometry of Manifolds, which helped modernize the teaching of geometry in the 1960s.
- Barrett O’Neill: Their joint work on warped products remains one of the most cited papers in the intersection of geometry and physics.
- Isadore Singer: As a student of Singer, Bishop was part of an intellectual lineage that includes some of the most influential geometers of the 20th century.
7. Lesser-Known Facts
- The "Natural" Frame: Bishop’s 1975 paper "There is more than one way to frame a curve" was somewhat ignored by pure mathematicians at first but was later "rediscovered" by computer scientists. It is now a standard algorithm in 3D modeling software for creating "tubular" surfaces (like hoses or wires) without the surface twisting unnaturally.
- A Passion for the Outdoors: Bishop was an avid hiker and nature lover. Colleagues often noted that his intuition for "space" and "curvature" seemed to be informed by his physical experience in the natural world.
- Longevity at Illinois: In an era where top-tier mathematicians often moved between elite institutions, Bishop’s 60-year association with the University of Illinois (from 1959 until his death in 2019) was a testament to his loyalty and the strength of the UIUC mathematics department during that era.