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11 of 12 people found the following review helpful
5.0 out of 5 stars A practical elementary introduction to classical differential geometry, May 30, 2009
This review is from: Schaum's Outline of Differential Geometry (Schaum's) (Paperback)
After so many years, this book continues to be a valuable introduction to the differential geometry (DG) of curves and surfaces in the euclidean 3-dimensional space R^3, quite clear and efficient for self study, since each chapter combines a serious bulk of theory and many solved exercises, as well as some unsolved problems. It starts reviewing much of the necessary calculus needed. Then, it goes into curves, defining curvature and torsion, and proving the Frenet-Serret equations. It is shown that every regular curve is detrmined by its curvature and torsion (up to a rigid motion). Many interesting problems on curves illustrate the theory. But little attention is given to plane curves and no global property of curves is given (what is quite understandable, since they are hard to prove). The book continues with surfaces, defining parametrizations, atlas, the tangent plane and the differential of a map of surfaces. Then, we find an excellent introductory exposition of lines of curvature and assymptotic lines, (including Meusnier, Euler, Rodrigues and Beltrami-Enneper theorems) as well as geodesic curvature, geodesics, mean and Gauss curvature. The so called fundamental existence and unicity theorems for curves and surfaces in R^3 are stated and proved, as well as Gauss Theorema Egregium. However, there is no mention of parallel transport (you can find this in Stoker Differential Geometry (Wiley Classics Library), in Goetz Introduction to Differential Geometry (Addison-Wesley Series in Mathematics), in Millman-Parker Elements of Differential Geometry's, in do Carmo's Differential Geometry of Curves and Surfaces or in Klingenberg's A Course in Differential Geometry (Graduate Texts in Mathematics), all of them introductory books on DG too. The book also treats the simplest global properties of surfaces: (1) orientability (mildly presented), (2) Liebmann's theorem characterising compact connected surfaces of constant curvature in R^3 as spheres ( clearly proved, without assuming its orientabilty), (3) Gauss-Bonnet theorem, proved in a rather sketchy way, but well illustrated in some exercises, which clarify its meaning and difficulty. In general, many theoretical properties are proved as exercises. Practical questions are easy or not too hard to solve. If you really don't know the subject, this book is a perfect start, alone or combined with those previously cited works, or with Struik's classical Lectures on Classical Differential Geometry: Second Edition, Oprea's Differential Geometry and its Applications (Classroom Resource Materials) (Mathematical Association of America Textbooks), or Montiel-Ros' Curves and Surfaces (Graduate Studies in Mathematics). Other problem books on the DG of curves and surfaces are rare. I will mention (1) Fedenko's (Mir Editions, now re-edited by USSR (sic!) editions-Moscow) (similar to M. Lipschutz's level, but much less detailed and with no theory). (2) Mishchenko-Solovyev-Fomenko (Problems in DG and Topology, Mir- Moscow).
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