This chapter has a topological flavor, but the topological prerequisites (connectedness and compactness in Euclidean spaces) are largely summarized in an appendix to the chapter.Īs the above summary of the contents of this book should make clear, there is a lot of material covered in this text, far more than can be covered in a single semester. This rather lengthy chapter is divided into eleven subsections, many independent of the others, each proving a “big” theorem in the subject for example, the Hopf-Rinow theorem on geodesics. The final chapter of the book is on global differential geometry, both of the surface and curves in three-space. Highlights of this chapter are the Theorema Egregium of Gauss (good luck finding this in the Index, though after fruitless searches under T and E, I finally found it under G for “Gauss theorem Egregium”) and the Gauss-Bonnet theorem, as well as the subjects of parallel transport and the covariant derivative. Another way to put this is that intrinsic geometry explores concepts that can be determined from an understanding of the first fundamental form. Intrinsic properties of a surface are those that depend only on the surface itself and do not depend on the way it is situated in the ambient Euclidean space. The intrinsic geometry of a surface is addressed in the next chapter. This chapter explores these, introduces the second fundamental form, and ends with a few optional sections in which, among other things, ruled surfaces and minimal surfaces are discussed. The derivative of the Gauss is not just any ordinary linear mapping it turns out to be self-adjoint, and many of its algebraic features (eigenvalues, determinant, etc.) have geometric significance. This chapter also introduces the first fundamental form, a quadratic form that can be used to compute a lot of information about the surface (length and angle, for example).Ĭhapter 3 introduces the Gauss map and explores its properties and applications. The definition of “regular surface” given in the second chapter is one that generalizes easily to manifolds, and much later in the text the author does discuss the notion of a differentiable manifold, first discussing “abstract surfaces” (i.e., 2-dimensional manifolds) and then noting that there was really nothing special about the number 2 and that the idea generalizes to n-dimensions. The local theory addresses things like the Frenet equations, while the global theory discusses curves “in the large”, including theorems like the Four-Vertex theorem and isoperimetric inequality. The chapter on curve discusses both the local and (for plane curves) global theory. The first two cover curves and surfaces, respectively, in three-space (and sometimes in the plane). Most books with titles like this offer similar content. Yet, there must still be some market for books like this, because several have recently appeared, including a second edition of Differential Geometry of Curves and Surfaces by Banchoff and Lovett and another book with the same title by Kristopher Tapp. Of late, however, it seems to me (based on anecdotal evidence garnered from a highly unscientific survey) that not as many departments offer such a course. I now find myself in the position of once again thanking them, this time for publishing not just a re-issuance, but an actual new edition, of do Carmo’s classic textbook, first published by Prentice-Hall (now part of Pearson) in 1976.īack in the day, it was fairly common for undergraduate mathematics departments to offer a course in differential geometry, which I suppose I should now refer to as “classical” differential geometry (curves and surfaces in the plane and three-space) to distinguish it from “modern” differential geometry (the study of differentiable manifolds). For some years now, I, as well as a number of other contributors to this column, have on occasion expressed appreciation to Dover Publications for the service it provides to the mathematical community by re-issuing classic textbooks and making them available to a new generation at an affordable price.
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