Geometry

Our sources of geometric thought and writing go back to ancient civilizations, most notably the Greek and the Chinese. As an introduction to this vast field, we recommend the book by Hilbert and Cohn-Vossen [1], which illustrates the variety of geometric thinking within mathematics. We group our recommendations into six subareas.

Elementary Geometry.   This field studies simple geometric figures and their relationships. We mention Coxeter and Greitzer [2] as an excellent source. We also recommend the book by Pedoe [3], which contains a wealth of material on circles and spheres.

Discrete Geometry.   This area in geometry is dominated by the Hungarian school of thought, which includes work on packings and coverings as studied by Laszlo Fejes-Toth and combinatorial extremum problems as popularized by Paul Erdos. We recommend the text by Pach and Agarwal [4], which discusses a broad range of problems and results in the area.

Convex Geometry.   Convex sets have been studied from a variety of angles. A big topic in this field is convex polytopes, and we recommend the relatively recent text by Ziegler [5]. Other topics include cross-sections and projections of convex bodies and their measurement, and we recommend the book by Schneider [6] as an introduction to that literature.

Stochastic Geometry.   The combinatorial viewpoint in discrete geometry is related to, but not the same as the probabilistic focus we find in stochastic geometry. A fairly recent book on the topic is Klain and Rota [7]. A related area is the study of shape spaces understood as the set of deformations of a collection of measurements. We mention the text by Kendall [8] as an introduction to this school of thought.

Differential Geometry.   This is probably the largest subarea in geometry. At the more elementary end of the spectrum it includes vector calculus, and at the other end it connects to Riemannian geometry and other advanced topics. We recommend the text by O'Neill [9] for a general introduction, and the book by Morgan [10] for an introduction to Riemannian geometry.

Singularity Theory.   This is another advanced topic in differential geometry and concerns itself with the classification of metamorphisms that occur in the generic deformation of smooth shapes. A delightful text in this field is the book by Bruce and Giblin [11], which studies curves and surfaces. Catastrophe theory is a synonymous name for the field and also the title of an introductory booklet by Arnol'd [12].


References:
[1] D. Hilbert and S. Cohn-Vossen. Geometry and the Imagination. AMS Chelsea, Providence RI, 1990.
[2] H. S. M. Coxeter and S. L. Greitzer. Geometry Revisited. Mathematical Association of America, Washington DC, 1967.
[3] D. Pedoe. Geometry: A Comprehensive Course. Dover, New York, 1988.
[4] J. Pach and P. Agarwal. Combinatorial Geometry. Wiley, New York, 1995.
[5] G. Ziegler. Lectures on Polytopes. Springer-Verlag, New York, 1995.
[6] R. Schneider. Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, England, 1993.
[7] D. A. Klain and G.-C. Rota. Introduction to Geometric Probability. Cambridge University Press, England, 1997.
[8] D. G. Kendall. Shape and Shape Theory. Wiley, Chichester, England, 1999.
[9] B. O'Neill. Elementary Differential Geometry. Academic Press, San Diego CA, 1997.
[10] F. Morgan. Riemannian Geometry. Jones and Bartlett, Boston, 1993.
[11] J. W. Bruce and P.J. Giblin. Curves and Singularities. Cambridge University Press, England, 1982.
[12] V. I. Arnol'd. Catastrophe Theory. Springer-Verlag, Berlin, 1992.