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Book review

Author(s) Hermann, R.
Title Cartan Geometry, Nonlinear waves, and Control Theory, Part B
Publisher Math Sci Press
Year of publication 1980
Reviewed by M. Marinov

This is one of the volumes in the series composed by Robert Hermann and devoted to applications of geometrical methods in theoretical physics. Physics and geometry have a long and interesting history of mutual enrichment. Newton's celestial mechanics was based essentially upon methods derived from classical geometry of the Hellenic era. In the next century, Lagrange was proud that his exposition of analytical mechanics does not need any figures. A century later, Poincaré introduced advanced geometrical ideas into celestial mechanics and, by the way, put foundation to topology. Einstein's general relativity was a triumph of geometrical ideas in physics. Geometry has got the new significance in physics during the past few decades, since the advent of non-Abelian gauge field theories. Now the Lie groups, the Cartan geometry and differential forms, topology of fibre bounds became standard tools of theoretical physics.

The series prepared by Hermann appeared as a response to the new trends in geometrization of physics. The volume consists of five parts (called Groups A-E). Group A, "Reflections" contains an emotional presentation of the author's view on the progress of mathematical physics, including his personal recollections. The titles of Groups B-D are self-explanatory. "B. Lie theoretic and differential-geometric foundations of the theory of stochastic processes and control". A discussion of relations between the theory of stochastic processes and quantum mechanics is also included here. "C. Geometrical methods in the theory of nonlinear differential equations". This is the most informative part of the book, including presentations of inverse scattering methods, Dikii-Gelfand theory of pseudodifferential operators, Hamiltonian dynamics, sine-Gordon equation, symplectic foliations etc. "D. Geometric system theory". Here one can find a theory of global linear systems, a geometric interpretation of input systems and Pfaffian systems. The last (but not the least interesting) Group E contains an English translation by M. Ackerman of two works by Sophus Lie: "General investigations of differential equations which admit a finite continuous group" and "Foundations of the theory of infinite continuous transformation groups".

The book contains a lot of various information presented in an original manner, even though not always organized in the best way. In Group A a reader would miss the second list of references (probably, to works by the author himself). The volume was prepared still in the pre-TEX era from camera-ready manuscipts, so the printing looks somewhat archaic now. The density of information per unit area is lower than in modern texts. One may hardly recommend the book for individuals, but no professional library of physics and mathematics would be complete without the Hermann series.