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Semi-Riemannian Geometry With Applications to Relativity, 103, Volume 103 (Pure and Applied Mathematics) [Hardcover]

by Barrett O'Neill
4.2 out of 5 stars  See all reviews (10 customer reviews)

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Book Description

July 12, 1983 0125267401 978-0125267403 1
This book is an exposition of semi-Riemannian geometry (also called pseudo-Riemannian geometry)--the study of a smooth manifold furnished with a metric tensor of arbitrary signature. The principal special cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry. For many years these two geometries have developed almost independently: Riemannian geometry reformulated in coordinate-free fashion and directed toward global problems, Lorentz geometry in classical tensor notation devoted to general relativity. More recently, this divergence has been reversed as physicists, turning increasingly toward invariant methods, have produced results of compelling mathematical interest.

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Editorial Reviews

About the Author

Barrett O'Neill is currently a Professor in the Department of Mathematics at the University of California, Los Angeles. He has written two other books in advanced mathematics.

Product Details

  • Series: Pure and Applied Mathematics (Book 103)
  • Hardcover: 468 pages
  • Publisher: Academic Press; 1 edition (July 12, 1983)
  • Language: English
  • ISBN-10: 0125267401
  • ISBN-13: 978-0125267403
  • Product Dimensions: 9 x 6.4 x 1.2 inches
  • Shipping Weight: 1.7 pounds (View shipping rates and policies)
  • Average Customer Review: 4.2 out of 5 stars  See all reviews (10 customer reviews)
  • Amazon Best Sellers Rank: #468,499 in Books (See Top 100 in Books)

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Customer Reviews

4.2 out of 5 stars
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Most Helpful Customer Reviews
26 of 27 people found the following review helpful
5.0 out of 5 stars The Best Introduction to General Relativity June 3, 2006
If you want to engage in a serious study of general relativity, then you must master the mathematical language of semi-Riemannian manifolds in which it is cast. Sadly, the development of classical Riemannian geometry as studied by pure mathematicians only parallels the development of semi-Riemannian geometry in the early stages; eventually, the two subjects diverge rather drastically. For example, the famous Hopf-Rinow Theorem, one of the cornerstones of modern Riemannian geometry, simply has no Lorentzian analogue at all; every single equivalence in the theorem fails in Lorentzian geometry. Thus, one could master all five volumes of Spivak's definitive treatment of Riemannian geometry and still be unprepared to deal with light cones, timelike, null and spacelike geodesics, and the multitude of other uniquely semi-Riemannian constructs that appear in general relativity.

O'Neill's wonderful book, which first appeared in 1983, provides the well-prepared reader with a mathematically rigorous, thorough introduction to both Riemannian and semi-Riemannian geometry, showing how they are similar and pointing out clearly where they differ. After developing the mathematical machinery in the early chapters, the last part of the book turns to general relativity by offering lucid introductions to the Robertson-Walker cosmological models (Big Bang singularities), the Schwarzschild model for a single non-rotating star (including black holes), and a brief introduction to Penrose-Hawking causality theory.

If you would like to study a pure Riemannian text in parallel with this one, I would recommend the text by Boothby, written at a comparable level of difficulty, which remains one of the clearest and most accessible Riemannian geometry texts on the market.
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14 of 15 people found the following review helpful
3.0 out of 5 stars Great book, terrible print quality February 18, 2006
This is a wonderful book, with a clear, concise and precise exposition of the fundamental idea in riemannian and semi-riemannian geometry. Although I would not recommend it as a first text, it will be the text that you continue to reference later, and turn to when you want the best mathematical treatment.

However, I do not recommend that you buy a new copy. The print quality is terrible; the binding is poor, but even worse, the text quality is absurd. I have been using a library copy with cloth binding and sharp, clear text. It is obvious that the new printing in the green cover is based on a photocopy of the original rather than a new typesetting. While this means that no errors have been introduced, I found it painful to read. I would suggest looking for a used copy.

So 5 stars for the book, but only 3 stars for this printing.
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9 of 12 people found the following review helpful
This is one of the best books on Differential Geometry I've ever read. It includes a clear exposition of all the basic results and then goes on to the most deep aspects of the subject, making it useful for undergraduate and graduate students, as well as experienced working mathematicians. It's a pitty that it's no longer available.
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2 of 2 people found the following review helpful
Format:Hardcover|Verified Purchase
Professor O'Neill was my teacher and PhD supervisor. He was the best teacher I or anyone else ever had. First of all he knew the material as it should be known, namely through clean coordinate-free geometric understanding; afterwards the coordinates can be chosen when specific local calculations are needed. Secondly, he was able to assemble the material and express it cleanly and clearly.

Everybody should know the basics of this material, so that the geometric content of the Calculus will finally become common knowledge. This knowledge is beautiful, powerful, clean and simple. (I mean you professional mathematicians and pedagogues across the world, as well as students and all people who need to use mathematics to solve problems, should know these basics. They were known to the classical mathematicians, even if they could not make it explicit or communicate it directly.)

Modern-day pedagogues are still writing and teaching all about coordinates even when they have nothing to do with the discussion at hand. Mathematics needs to catch up to where it was in the past.
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5.0 out of 5 stars A must-read for anyone working in relativity. February 8, 2014
By Bartosz
The best book on semi-Riemannian geometry out there. A must-read for physicists and mathematicians interested in relativity. Many mathematics texts on smooth manifolds cover only Riemannian geometry.
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