The book under review is only the first part of a projected pair: the present volume deals with what the authors characterize as “three building blocks, each consisting of four chapters,” and covering, respectively, calculus on manifolds, Lie theory (including Lie group actions) and symplectic geometry, and finite dimensional Hamiltonian systems. The presence of the latter as something of a culmination of the discussion points to the objective behind the book, namely, to get to the formal structure of modern physics in a thorough and effective fashion.
This physicist’s point of view is amplified by the authors’ explicit focus on “the concepts of symmetry and integrability and on Hamilton-Jacobi theory.” Under this umbrella we encounter some true gems, such as the Arnol’d conjecture (p. 487 ff.), non-commutative integrability (p. 627 ff.), and Morse families (p. 658 ff.). The latter are due to Lars Hörmander. Having been introduced in his hugely famous 1971 Acta paper, “Fourier integral operators, I,” and christened “phase functions” by him, they fit into his theory of oscillatory integrals and accordingly evince deep connections to Feynman’s version of quantum mechanics and electrodynamics as well as symplectic geometry. In the present treatment they are characterized as generalizations of Morse functions (cf. the excellent discussion of Morse theory in § 8.9 of the book), and their power is illustrated by, e.g., Corollary 12.4.10 (p. 668) to the effect that “[l]ocally, every Lagrangian immersion [of a Lagrangian submanifold in a cotangent bundle] is generated by a Morse family.” For the definition of a Morse family, as such, the reader should consult Definition 12.4.2 (p. 660).
Indeed, then, this is a book well worth reading — carefully. It is full of very exciting differential geometry, and beautiful and deep physics to boot. And it is a marvelous thing to learn differential geometry in the context of its role in modern physics: just consider the historical bombshell of Einstein’s general relativity and its dependence on Riemannian geometry as well as the presence of symplectic geometry in quantum mechanics (it arises very naturally from the non-commutative quantization relations).
By the way, if you’re worried about this book being too much in the way of physics, just consider that neither Einstein nor Heisenberg, Schrödinger, or Dirac appears in the index: this really is a mathematics text!
And it’s huge: over 700 pages. But it’s well-written and scholarly, comes equipped with remarks and exercises, and is pitched at a good level. Additionally, as far as differential geometry proper goes, there is a great deal of wonderful stuff present, from de Rham cohomology and some Hodge theory to (a lot of) symplectic geometry, as already indicated. The authors’ discussion of Lie theory is also very nice and complete: two central chapters are devoted to this important material. To lift Feynman’s phrase to the present mathematical context: “a lot of the good stuff.”
This very good book makes one impatient for the appearance of the second volume!
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.