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Tensors and Riemannian Geometry

Nail H. Ibragimov
Walter de Gruyter
Publication Date: 
Number of Pages: 
De Gruyter Graduate
[Reviewed by
Michael Berg
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Tensors are in some ways contested things: mathematicians and physicists, both of whom use them with tremendous frequency, tend to flavor them rather differently. For me, who first encountered them in an unabashedly algebraic context, they will always be objects that make the multilinear linear, so to speak: the tensor product of appropriate algebraic structures, such as vector spaces or modules over a ring, enjoys a universal property that says that any n-multilinear mapping from an n-fold Cartesian product of such structures to a suitable target structure, factors through the corresponding n-fold tensor product to produce a linear mapping. Fair enough, but how do we compute with n-fold tensors? Well, there is of course the equivalence relation perspective that permits one to “mod out” by n-multilinearity: just descend to the quotient by the multilinearity relations (or the sub-object generated by them) and play a back-and-forth game accordingly. In any case it’s all very comfortable algebraic stuff.

Well, not so much: tensors are among the most cherished tools of the differential geometers and therefore, in the wake of Einstein, the physicists. I guess I first saw what physicists do along these lines in Dirac’s book on general relativity, and was struck by a moral similarity with what he had done in his Principles of Quantum Mechanics with dual bases for vector spaces: notations were introduced to facilitate austere calculations that are unavoidable in physics, but that really flavor these objects very differently. And it is the case that these conventions are prevalent, too, in Riemannian and even differential geometry in general, given that, after all, these areas are most intimately connected to the physics of the universe.

The book under review is flavored with the latter ingredients; physics pervades the discussion. The introduction of the topic of tensors and Riemannian spaces in Chapter 3, for example, is preceded by Chapter 2 which is devoted to conservation laws, and the substantive discussion of second order differential equations in the middle chapters of the book is followed by a sizable treatment (in fact, the entire Part III of the book) of relativity. That said, Ibragimov devotes a lot of space, and appropriately so, to Riemannian geometry as such, as well as to the minutiae of the various differential equations that come to bear on the matter.

Happily, in his chapter on conservation laws, there is an explicit discussion of Noether’s (physics) theorem, as well as its context, the Euler-Lagrange equations. Beyond this, the Cauchy problem figures very prominently in his discussion of differential equations, while Ibragimov’s (long) treatment of relativity seems to cover just about everything. In the latter, he goes from special relativity, the Lorentz group, and Maxwell’s equations to general relativity, including a discussion of Mercury’s parallax. The book’s final chapter is titled “Relativity in de Sitter space” and gets to such specifics as particle motion and the behavior of neutrinos: Dirac’s equation (first introduced in the preceding chapter) figures prominently here. The very last chapter of the book is particularly tantalizing: “Splitting of neutrinos by curvature.” It doesn’t get any more modern than that, as far as physics (or cosmology) goes.

So, Tensors and Riemannian Geometry is an important text book (yes: it has exercises galore) on many counts, both qua geometry and qua physics, and is geared to provide the reader (or, rather, the problem worker) with real facility in the areas covered. It is a very important and useful contribution to the literature.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.