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Differentiable Manifolds: A Theoretical Physics Approach

Gerardo F. Torres del Castillo
Publication Date: 
Number of Pages: 
[Reviewed by
William Porter
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Differentiable Manifolds: A Theoretical Physics Approach is an introductory text on its namesake subject written for advanced undergraduate or early graduate study. After beginning with the fundamental definitions, the author quickly demonstrates that his approach is indeed geared toward those with more physically-oriented interests. Within the first twenty pages, the author defines contravariant vectors, mentions the Lagrangian and its connection to the tangent bundle, and includes an early example of the Lie bracket as an operation on vector fields.

The author then goes on to treat the algebra of forms, Lie derivatives, exterior derivatives, and to introduce the idea of a connection, with which come the standard computations and exercises associated with torsion and curvature. Curvature and the metric tensor are treated in great detail. Then, toward the end of the book, there are applications to Hamiltonian mechanics, geometric optics, and Euler’s equations for rotational dynamics.

I’m not sure that this well-written book is really written for the audience to whom it is advertised. The level of mathematical rigor is well beyond anything with which most undergraduate or first-year graduate students in physics are comfortable. Even though the prerequisites are reasonable, the pace and attention to detail will tax students without training in formal proof and argumentation. Mathematics students may not find the physical examples motivating, and they probably will not have experience with the principles underlying analytical mechanics and geometric optics, let alone the Schwartchild solution.

Furthermore, much of this development is done without the aid of a geometric interpretation, which I think is very detrimental to the quality of understanding that this book can provide.

I would recommend this book to someone with a background in both physics and mathematics, but I cannot say that I would recommend this text to someone with a background in either subject alone. If choosing an introductory text, I would probably look elsewhere. I applaud the author’s ability to cover so much in a way that is seldom seen in formal coursework, and I personally have been looking for this type of book for the last couple of years. In the future, I will consult this text to look for connections between physical systems and the mathematics of differentiable manifolds, but I see myself looking to other references for understanding thereof.

William Porter has a B.A. in mathematics-physics from a small liberal arts college and works as a systems engineer. He enjoys ballroom dancing, cooking, and T. S. Eliot’s poetry. He thinks that Rachmaninoff writes amazing music and rain is the best weather. His favorite Big Bang Theory character is Sheldon.