Review by Choice Review
Differential geometry is a vital field, important to both mathematics and physics. It is a pity that it is not always a regular part of the curriculum, since its primary elements are accessible to undergraduates with backgrounds in linear algebra and some multivariable analysis. Bar (Univ. of Potsdam, Germany) begins this work with an optional initial chapter concerning Hilbert's axioms for Euclidean geometry. He then treats classical curve and surface theories, including global results as Fenchel's theorem on total curvature of space curves and the Fary-Milnor theorem on knotted curves. Following these discussions, Bar addresses some of the intrinsic geometry of surfaces. Here, the author develops the heavy machinery of Riemannian curvature, geodesics, parallel transport, etc., to give background for the final chapter, which proves the Gauss-Bonnet theorem. Along the way, Bar includes a lovely discussion of cartography and demonstrates different models for hyperbolic geometry via Escher-like interpretations. Notable is Bar's elegant progression from elementary calculations to more intricate ones. Exercises are peppered throughout the text; about a hundred (unfortunately) significant hints are provided for almost all of them at the end of the work. Nonetheless, Bar's book is a welcome addition to the literature. Summing Up: Highly recommended. Upper-division undergraduates and graduate students. S. J. Colley Oberlin College
Copyright American Library Association, used with permission.
Review by Choice Review
