Review by Stephen Parrott 

                                        of

			Curvature in Mathematics and Physics

					by

				Shlomo Sternberg

                                   Dover, 2012
			

	Although billed as a text for undergraduates and beginning graduate students, only the best-prepared and exceptional undergraduates are likely to get much out of this book. More realistic preparation would be a good course in modern differential geometry.

	The book uses the modern definition of "differential manifold" throughout, but I can't find it defined anywhere in the book. The grossly inadequate index contains only 17 items starting with "m" , and these do not include "manifold" ! The closest to a definition seems to be a definition of "parametrized surface" in Chapter 1.

	I doubt that anyone without a previous acquaintance with differential forms will get much out of this book. Technically, the definitions are given in Chapter 2, but in only a few pages in a very abstract way. Some of that treatment seemed to me downright perverse. For example, the concept of "Lie derivative" of a vector field has a simple geometrical interpretation given in almost all texts. From this follows the related concept of Lie derivative of a differential form field, whose geometrical motivation depends on the previous concept of Lie derivative of a vector field. But Sternberg introduces these concepts "backward" starting with a geometrically unmotivated algebraic definition of Lie derivative of a form field (which relies on the nontrivial Weil formula) and from that produces an algebraic definition of Lie derivative of a vector field. I would be surprised if anyone unfamiliar with the general concept of "Lie derivative" will come away from this discussion with an adequate understanding.

	That said, there is a lot of value in this book for those already familiar with differential geometry. I enjoyed browsing through it. It includes many pictures of prominent differential geometers, with some biographical sketches. Most of the mathematics is presented at an appropriate level of rigor, perhaps not quite as high as mathematical research, but much higher than typical physics literature. Though the exposition is more informal than that of most mathematics texts, it is usually no less rigorous.

	Apart from applications to general relativity, most of the physical applications were presented so sketchily that they were hard for me to follow in detail. The "Curvature in ... Physics" part of the title should be taken with a grain of salt by readers without extensive experience in the relevant fields of physics.

	I cannot recommend this book as a textbook, but I have enjoyed browsing through it and am happy to have it in my library. I think that for its price, it is a bargain.

Stephen Parrott

July 25, 2016