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Curvature in Mathematics and Physics

Shlomo Sternberg
Dover Publications
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
P. N. Ruane
, on

Within the 400 pages of this Dover original publication, there are 21 photographs that humanize its mathematical contents. These are portraits of the most notable mathematicians and physicists whose work has led to the formulation of modern differential geometry and its applications to relativity, electromagnetic theory, particle physics and geometry.

Because Riemann’s ideas underpin the vast edifice of contemporary differential geometry, his face is the first to appear. We next catch a glimpse of Albert Einstein, to whom Riemannian geometry was such an indispensable tool. Of course, without the exterior differential calculus, the methods of moving frames and symmetric spaces etc, a book like this could not have been written. Hence the inclusion of a rather blurred image of Élie Cartan.

The fourth photograph is that of Carl Friedrich Gauss (it’s actually a photograph of a painting). So chapter 1 begins by explaining how curvature is ascertained by calculating the volume of a hypersurface thickened on one side. Gauss’ theorema egregium arises from a situation in which a 2-d surface is evenly thickened on both sides. In other words, the integrands are determined by the intrinsic geometry of the surface, and not the ambient space. Six chapters later, Gauss’ lemma is used to show that geodesics locally minimize length up to a conjugate point on any semi-Riemannian manifold.

Following an early introduction to the various curvatures of hypersurfaces, there is a short review of calculus on manifolds. The importance of linear connections and their curvatures is then described, and the reader is introduced to Tullio Levi-Civita the 6th (photo no.5 depicts the creator of the Weingarten map). At this stage, the Levi-Civita theorem leads to methods for calculating geodesics in the exterior of the Schwarzschild black hole. I’d never heard of Schwarzschild before — which makes me think that I haven’t been paying attention, because he provided the first exact solution of a limited case of the Einstein field equations.

A brief summary of the remaining mathematical contents of the book would say that, after the Levi-Civita theorem, there is discussion of bivariant metrics on Lie groups, Cartan calculations, and variational formulae (alack, alas, no picture of Sophus Lie!). Then, with cigarette in hand, Heinz Hopf meets up with Willi Rinow to tell us that M is complete as a metric space iff it is geodesically complete. A hundred pages later on, a bearded and bow-tied Frobenius has a chapter devoted to vector fields of a differential system.

Smoothly embedded within the mathematical contents of this book is a range of applications that exemplify the achievements of 20th century mathematical physics. For example, the set problems on geodesics (in the Schwarzschild exterior) refer to massive and massless particles, orbit types and perihelion advance. Chapter 11 provides a review of special relativity, and there is subsequent consideration of gauge theories and the Higgs mechanism in the context of the reduction of principal bundles. Also, appearing within a chapter on Minkowski space, there is a very accessible interpretation of Karl Compton’s formula. I think that this basically implies that, when a photon impinges on a static massive particle, a shift in wavelength is independent of the wavelength of incoming radiation.

Going back a bit, chapter 12 considers the importance of the [Hodge] star operator, which facilitates the comparison of Maxwell’s equations (for electromagnetism) with the London equations (for superconductivity). Yikes! I always thought the London equations were eponymously British; but the 16th photograph in Shlomo Sternberg’s ‘pictures in an exposition’ shows the London brothers Fritz and Heinz at Cambridge in 1953; and I now know that they were born in Bonn, Germany just before WW1. (Judging by the picture, they look more like father and son!)

In summary, I wish I’d had this book 40 years ago when, as a postgraduate student, I was trying to comprehend some of the relativistic aspects of Maxwell’s equations. But the present reason for recommending Shlomo Sternberg’s marvellous book is that it provides an inspiring overview of contemporary differential geometry and its varied applications. What’s more, this highly imaginative text is priced at the amazingly low price of $19.95.

Peter Ruane is retired from a career spent in the provision of courses for teachers of (primary and secondary) mathematics.

1. Gauss's Theorem Egregium
2. Rules of Calculus
3. Connections on the Tangent Bundle
4. Levi-Civita's Theorem
5. Bi-invariant Metrics on a Lie Group
6. Cartan Calculations
7. Gauss's Lemma
8. Variational Formulas
9. The Hopf-Rinow Theorem
10. Curvature, Distance and Volume
11.Review of Special Relativity
12. The Star Operator and Electromagnetism
13. Preliminaries to the Einstein Equation
14. Die Grundlagen der Physik
15. The Frobenius Theorem
16. Connections on Principal Bundles
17. Reduction of Principal Bundles
18. Superconnections
19. Semi-Riemannian Submersions