# A New Approach to Differential Geometry using Clifford's Geometric Algebra by John Snygg.

BOOK: A New Approach to Differential Geometry using Clifford's Geometric Algebra by John Snygg, Birkhauser Boston, 2012.

REVIEWER: Izu Vaisman, Department of Mathematics, University of Haifa, Haifa, 31905, Israel (vaisman@math.haifa.ac.il)

The book under review is an interesting book for several reasons. The author's declared aim was to propose an alternative to the classical textbooks on Differential Geometry for undergraduate and beginner graduate students. In his textbook the algebraic instrument used in the study of geometry is Clifford's algebra, which replaces the usual vector algebra. More exactly, while assuming that the required vector spaces (essentially, the tangent spaces of manifolds) are endowed with a non degenerate (not necessarily positive definite) metric, these vector spaces are extended to the corresponding Clifford algebra and the computations are with Clifford numbers rather than vectors.

Once we know that, the geometry part of the book's content is what we should expect, even though, not in the classical order. After a brief introduction, Chapters 2-4 are dedicated to the study of the Clifford algebra of Euclidean 3-dimensional space, Minkowski 4-dimensional space and pseudo-Euclidean n-dimensional space, and to the use of the Clifford numbers in the study of the geometry of such spaces (rotations, reflections, etc.). Chapter 5 studies the intrinsic geometry of n-dimensional submanifolds in pseudo-Euclidean m-dimensional space and Chapter 6 establishes the required machinery and gives the proof of the Gauss-Bonnet theorem. Finally, Chapter 7 is dedicated to various aspects of extrinsic geometry, such as Frenet formulas for curves, ruled and developable surfaces, the shape operator, principal curvatures and curvature lines, etc. Four brief Appendices treating the matrix representation of a Clifford algebra and Taylor series of the Copernic model and Kepler's orbits are added.

The book ends with an ample list of references (the following references should be included in a second edition: C. Chevalley, The construction and study of certain important algebras, Math. Soc. Japan, Tokyo, 1955 and P. K. Rasevkii, The theory of spinors (in Russian), Uspehi Mat. Nauk, 10 (1955), 3-110).

The pedagogy of the book is interesting as well. Usually, subjects are first treated in the case of an example, then explained in the general case. The author sees computation of geometric invariants (curvature, etc.) as an important task and dedicates a lot of efforts to the computation methods and to concrete computations of examples, which enhances the student's feeling of finality of his effort. The student's motivation is further stimulated by the inclusion of several subjects of physics treated by differential geometry. Subjects of special relativity are treated in the chapter on Minkowski space, and subjects of general relativity appear in the chapters on curved spaces and include Einstein's equation, the Schwarzschild metric, the precession of Mercury, the bending of light, etc.

Let us recall again that Clifford numbers, not vectors, are used in the development of every subject. Not always does this make a big technical difference but, it sheds more light on the subject.

Finally, the book is particularly interesting because the author places the development of differential geometry in real life. Mathematics and physics were developed by people who lived in different historical times, under specific political and economic conditions. The author includes in the book a large number of stories that tell us who these people were and describe their live and work. By these stories the reader learns about ancient Greek and middle-age Muslim mathematicians, he learns the (not just mathematical) biographies of Euler, Gauss, Cartan, Einstein, etc., he learns about German mathematicians during the first world war with its chemical warfare, he learns about the life (and death) of Soviet outstanding theoretical physicists in Stalin's terror time, and so on. The historical material covers, approximately, a quarter of the book and adds freshness and color to the exposed mathematical theories. Of course, these parts of the book (marked by an asterisk) will be recommended as free reading not as a part of the mathematical curriculum.

My guess is that not too many instructors will decide to teach differential geometry following this textbook, because of possible difficulties their students might later have in communicating with the majority of teachers and colleagues, who use the classical approach to differential geometry, and in studying non-metric differential geometry. But, I warmly recommend the book as a supplementary reading for students and, even more, for advanced students and for the mathematical public familiar with the basics of Differential Geometry, since the effort of reading this book is well rewarded.
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