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

Barrett O'Neill (Author)

Book Description

ISBN-10: 0125267401 | ISBN-13: 978-0125267403 | Publication Date: July 12, 1983 | Edition: 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.

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

• Hardcover: 468 pages
• Publisher: Academic Press; 1 edition (July 12, 1983)
• Language: English
• ISBN-10: 0125267401
• ISBN-13: 978-0125267403
• Product Dimensions: 9.1 x 6.1 x 1 inches
• Shipping Weight: 1.7 pounds (View shipping rates and policies)
• Average Customer Review: 3.8 out of 5 stars  See all reviews (6 customer reviews)
• Amazon Best Sellers Rank: #433,859 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

19 of 20 people found the following review helpful:

5.0 out of 5 stars The Best Introduction to General Relativity, June 3, 2006

By 

A Reader - See all my reviews

This review is from: Semi-Riemannian Geometry With Applications to Relativity, 103, Volume 103 (Pure and Applied Mathematics) (Hardcover)

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. For the serious reader who wishes to continue on with a study of the Kerr solution to Einstein's equations, modeling the exterior spacetime of a rotating star, O'Neill wrote an entire book on the subject in 1995, now difficult to find but well worth tracking down. This 1995 text contains one of the clearest, most accessible introductions available to the difficult subjects of the algebraic classification of the Weyl curvature tensor and the corresponding Petrov classification of spacetimes.

I studied from O'Neill's 1983 text when it first came out and I have continued to use it as the primary text for an advanced undergraduate course I have taught over the past 20 years. It is not an "easy" text to read, but then, I have never found the "easy" introduction to differential geometry and general relativity. The reviewer who says this is not a suitable first text is simply in error; there is no better first text on the subject. If you have studied linear algebra, advanced calculus, and a little topology, then with dedication and hard work, you can learn more from O'Neill's text than from many of the far more popular recent texts, written by physicists, which attempt to circumvent the mathematics insofar as is possible while introducing general relativity. This is a perilous course for which the serious student will pay dearly later on, when she/he wants to study any of the many areas of modern physics in which differential geometry (differential forms, bundle theory, connections on a principle fiber bundle, gauge theory, etc.) plays an essential role.

11 of 12 people found the following review helpful:

3.0 out of 5 stars Great book, terrible print quality, February 18, 2006

By 

MathGradStudent - See all my reviews

This review is from: Semi-Riemannian Geometry With Applications to Relativity, 103, Volume 103 (Pure and Applied Mathematics) (Hardcover)

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.

9 of 12 people found the following review helpful:

5.0 out of 5 stars Excellent for beginner and experienced mathematicians, March 15, 2000

By 

Bernardo Vargas - See all my reviews

This review is from: Semi-Riemannian Geometry With Applications to Relativity, 103, Volume 103 (Pure and Applied Mathematics) (Hardcover)

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.