mathematics

Richard L. Bishop

Richard L. Bishop is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Richard L. Bishop rather than just read about it. In short: Richard Lawrence Bishop (August 12, 1931 – December 18, 2019) was an American mathematician who specialized in differential geometry and taught at the University of Illinois at Urbana–Champaign. Bishop went to Case Institute of Technology as an undergraduate, earning a B.S. in 1954.

Key takeaways

Richard L. Bishop belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Richard L. Bishop to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Richard L. Bishop from memory before moving on to harder problems.

Reference excerpt

Richard Lawrence Bishop (August 12, 1931 – December 18, 2019) was an American mathematician who specialized in differential geometry and taught at the University of Illinois at Urbana–Champaign. Bishop went to Case Institute of Technology as an undergraduate, earning a B.S. in 1954. Next he earned his Ph.D. from the Massachusetts Institute of Technology 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. At UIUC, his doctoral students included future UIUC colleague Stephanie B. Alexander. 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 fellows of the American Mathematical Society. The Bishop–Gromov inequality in Riemannian geometry, 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 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, 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. doi:10.2307/1995057 ; doi:10.1090/S0002-9947-1969-0251664-4 Richard L. Bishop. There is more than one way to frame a curve. Amer. Math. Monthly 82 (1975), 246–251. doi:10.2307/2319846 Richard L. Bishop and Richard J. Crittenden. Geometry of manifolds. Reprint of the 1964 original. AMS Chelsea Publishing, Providence, RI, 2001. xii+273 pp. ISBN 0-8218-2923-8; doi:10.1090/chel/344

References

Worked examples

Example 1 — a first encounter with Richard L. Bishop

Start with the simplest possible case. Write down what Richard L. Bishop claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Richard L. Bishop before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Richard L. Bishop ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Richard L. Bishop

In research: Richard L. Bishop appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Richard L. Bishop in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Richard L. Bishop is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1931 births, 2019 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Richard L. Bishop outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Richard L. Bishop in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Richard L. Bishop means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Richard L. Bishop out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Richard L. Bishop in simple terms?

Why does Richard L. Bishop matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Richard L. Bishop?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Richard L. Bishop.