# A Question about Theodore Frankel's "The Geometry of Physics"

Locked up in my self-distancing isolation in NYC, I'm reminded of how much I really like Frankel's book, which contains a wealth of beautiful geometry and topology from the standpoint of a mathematically minded physics student. It contains so much pure mathematics from a first year graduate course in differential geometry with so many pictures, but even more wonderful, it contains many applications of this material to physics, particularly classical mechanics and electromagnetism. Since I've always believed mathematics-REAL mathematics, with proofs!-and the physical sciences should never be separated, I naturally found the book an amazing find. Interestingly, as I've asked around, I've found that mathematicians and mathematics students generally like the book considerably more then the physicists the book was written for!

Here's my question: Despite it's beauty, as it was written for advanced physics students, it would be a bit problematic to use as a text for pure mathematics graduate students. While it certainly covers enough pure geometry for such a course, Frankel isn't as careful with his proofs as one would like for such a course. Indeed, he admits as much in his preface. I'd like to know what pure mathematics text would the geometers on the exchange would recommend as a supplement to "tighten up" such a course. Ideally, one wouldn't want that book to be too long and comprehensive, like Spivak, since it would repeat most of the material in Frankel. I was considering using Frank Warner's book. Ordinarily, I'd be horrified to recommend it as the sole text for a course, but as a supplement to Frankel, it might be ideal as the 2 books seem to complement each other beautifully! I'd love to try to use Conlon's beautiful book also, but doesn't seem to have enough range.

What do the geometers here think? Any other recommendations?

• I wish I had John Lee's and Jeffrey Lee's two masterful texts to complement Frankel when I read it. Certainly Frankel's text influenced me and it is a pretty book. springer.com/gp/book/9780387217529 and bookstore.ams.org/gsm-107 Mar 23 '20 at 14:14
• At a certain point in your second paragraph, you start referring to "such a course," but you don't say what the course is, except to say that its audience is "pure mathematics graduate students." Do you mean, for example, a course in differential geometry? Do you have any evidence that many math grad students want to take a course that aligns closely with this book's contents? From a quick glance through the table of contents on amazon's "look inside" feature, it seems that it's a thick, overpriced book, and a lot of the material is highly specialized.
– user507
Mar 23 '20 at 20:49
• @JamesS.Cook Totally agree and I recommend Frankel's book heartily to both mathematicians and physicists-if not as the main book for a course, then an important supplement. Mar 23 '20 at 21:37
• @BenCrowell Well,first of all, it covers most of the basics of any first year graduate course in differential geometry (that's the kind of course I was referring to), although as I said, there are a number of subtle results that don't have proofs, such as the Hopf-Rinow theorems. This would be expected in a book designed for physics students. Second, I know many mathematics students who are also interested in physics and vice versa. I'd love use it in a DG course with modern standards of rigor and yet serve both kinds of students.I've edited the question to make it clearer. Mar 24 '20 at 3:37
• Possibly worth a look, although none is really designed for pure mathematics graduate students: Manifolds, Tensor Analysis, and Applications by Abraham/Marsden/Ratiu AND Gauge Fields, Knots and Gravity by Baez/Javier AND Foundations of Mechanics by Ralph Abraham and Jerrold E. Marsden (1987 6th printing of 2nd edition freely available here, $\approx$ 89 MB). Mar 24 '20 at 14:59

## 2 Answers

I've owned the revised first edition of Frankel's The Geometry of Physics: an Introduction at least since I was a graduate student. The texts I suggest in this answer are largely based on my personal library.

## Part I: Manifolds, Tensors, and Exterior Forms

Contains 6 chapters which construct the canvas on which the later part of the text plays out. I'll sketch the content then follow with my recommended readings to complement Frankel:

• Chapter 1: Manifolds and Vector Fields (submanifolds, manifolds, construction of tangent vectors at a point, vector fields and flows in $$\mathbb{R}^n$$)
• Chapter 2: Tensors and Exterior Forms (covectors, the differential, differential forms, tangent and cotangent bundles, pull-backs, tensors, exterior algebra, how it works in $$\mathbb{R}^3$$, orientation, interior products and contraction)
• Chapter 3: Integration of Differential Forms (integration of p-forms in $$\mathbb{R}^n$$, line and surface integrals, manifolds with boundary, partitions of unity, inducing orientations on boundary, integration of psuedo-n-forms on n-manifold, Maxwell's Equations)
• Chapter 4: The Lie Derivative ( of vector fields, of forms, how to differentiate an integral, application to Hamiltonian mechanics )
• Chapter 5: Poincare Lemma and Potentials ( generalized Stokes' theorem, closed forms, exact forms, complex analysis, the converse to Poincare Lemma, finding potentials)
• Chapter 6: Holonomic and Nonholonomic Constraints (Frobenius integrability condition, distributions of vector fields and forms, the Frobenius Theorem, foliations and maximal leaves, systems of Mayer-Lie, holonomic and nonholonomic constraints, thermodynamics via Caratheodory)

Ok , let me pause with this partial recap of the table of contents to point towards a few things I'd recommend to partner with the above:

1. John M. Lee's Introduction to Smooth Manifolds is great for in-depth coverage of material in Chapters 1, 2, 3 and 4 and parts of 5 and 6.
2. Jeffrey M. Lee's Manifolds and Differential Geometry also covers nearly all the material in Chapters 1-6 of Frankel. I think Chapter 10 contains the nuts and bolts of the proof of Poincare's Lemma, which I think is an improvement on the sketch in Flander's Differential Forms with Application to the Physical Sciences ( incidentially, engineers at Purdue in the 1960's were apparently pretty awesome). Then Chapter 11 in Jeffrey Lee's text has a really nice uninterrupted study of the question and solution which is offered by Frobenius theorem. In my teaching, I found the differential forms and exterior product formula in this text to be a useful supplement where I got lazy.

## Part II: Geometry and Topology

This part of Frenkel is largely concerned with the addition of structure to manifolds which capture the concept of geometry in the Riemannian or semi-Riemannian sense.

• Chapter 7: $$\mathbb{R}^3$$ and Minkowski Space (Frenet frames in $$\mathbb{R}^3$$, 4-vectors and Minkowski Space, Electromagnetism in differential form notation on Minkowski Space)

• Chapter 8: Geometry of Surfaces in $$\mathbb{R}^3$$ (first and second fundamental forms, the Weingarten equations, principal curvatures, Gaussian curvature, mean curvature, Gauss map, Brouwer degree and fixed point theorem, Gauss-Bonnet Theorem, first variation of area, soap bubbles and minimal surfaces, Gauss's Theorem Egregium, geodesics, intrisic derivative, parallel displacement of Levi-Civita)

• Chapter 9: Covariant Differentiation and Curvature (covariant derivative or affine connection, coordinate frames, curvature of affine connection, torsion-free connections, the Riemann connection, Cartan's Exterior Covariant Differential, exterior covariant derivative of vector field or form, Cartan's structural equations, exterior covariant derivative of vector-valued form, curvature 2-forms, change of basis and gauge transformations, curvature forms in Riemannian manifolds, classical differential geometry ala Gauss recovered from Cartan's structure equation viewpoint, parallel displacement and curvature on surface, flat metrics, horizontal distributions, Riemann's theorem on flatness and construction of local frame)

• Chapter 10: Geodesics (vector fields along surface, geodesics, Hamilton's Principle in the Tangent Bundle, Hamilton's Principle in Phase Space, Jacobi's Principle of "Least" Action, closed geodesics and periodic motion, Geodesics Spiders and the Universe [aka detecting geometry from an intrinsic viewpoint]

• Chapter 11: Relativity, Tensors, and Curvature (calculus on curved space necessary for detailed understanding of Einstein's field equations, identities galore, Hilbert's variational approach to GR, curvature, sectional curvature, geometry of Einstein's equations, three dimensional version of Gauss's awesome theorem, remarks on Schwarzschild's solution)

• Chapter 12: Curvature and Topology: Synge's Theorem (second variation of arclength, Jacobi fields, conjugate points, Synge's Theorem stating closed geodesics are unstable in an even-dimensional orientiable manifold with positive sectional curvatures, also the application of Synge's Theorem towards simple connectivity as well as rigid body mechanics)

• Chapter 13: Betti Numbers and De Rahm's Theorem (singular chains and boundaries, singular homology groups, cycles and boundaries and homology and Betti numbers, homology groups of manifolds such as real projective space and torii, De Rahm's Theorem)

• Chapter 14: Harmonic Forms (the $$\ast$$ operator, scalar product on the exterior algebra, the codifferential operator, divergence in curved space, Maxwell's equations in curved space, Hilbert Lagrangian, Laplace operator on forms, harmonic forms on closed manifolds, Hodge's Theorem on solving Poisson's equation on a closed Riemannian manifold, Bochner's Theorem on vanishing Betti number, tangent and normal differential forms, Hodge's Theorem for Tangential Forms, existence of electric field subject to given boundary potential as special case of general result on harmonic field subject appropriate boundary conditions, relative homology, Hodge's Theorem for Normal Forms, Morse's Theory of Critical Points, Morse's Theorem)

And now for my recommendations. Following the order of the chapters,

1. Misner Thorne and Wheeler's Gravitation is still one of my favorites for reading about differential forms on Minkowski Space. Of course, there are a lot of books which help with Chapter 7 in Frenkel, I just have a certain nostalgia for MTW.
2. Barrett O'Neill's Elementary Differential Geometry revised 2nd edition does a great job of explaining the genesis of Cartan's method where you can actually see things. I like this book for a companion to Chapter 8 of Frenkel. Of course this material can be found many many other places.
3. Jeffrey Lee's text covers a lot of the same ground as Part II. I think it also does a good job of giving a more abstract treatment of what is done in O'Neill's more elementary text.
4. Gregory L. Naber's Topology, Geometry and Gauge Fields: Foundations discusses some of the topology touched on in Part II of Frenkel, but its real use is in providing more discussion of monopoles and their connection to geometry and topology. Also this has connections on fiber bundles in a later chapter with an aim towards application in physics.
5. Gregory L. Naber's Topology, Geometry and Gauge Fields: Interactions here he covers integration on manifolds, De Rahm cohomology and there is an appendix on Seiberg Witten theory which I am jealous to understand (my student is currently borrowing this text and I need to force him to teach it to me in repayment)
6. Shigeyuki Morita's Geometry of Differential Forms I probably should have this listed for Part I of Frenkel as well. This little book covers much and what I have read of it made a lot of sense. It has proofs of De Rahm's Theorem as well as the Hodge Decomposition. There is much to see here.
7. Lawrence Conlon's Differentiable Manifolds can be useful to get a quick big picture of some of the math in Part II of Frenkel. In particular, I recall the Cartan frame calculus is written at an interesting level in Conlon. Some attention is paid to the interplay between calculus on bundles and topology of the basespace.

## Part III: Lie Groups, Bundles, and Chern Forms

This part of Frenkel is largely concerned with the addition of structure to fiber bundles which capture the concept of local symmetry and thus lead to the natural formulation of Gauge theory on curves space

• Chapter 15: Lie Groups (Lie groups, invariant vector fields and forms, one parameter subgroups, Lie algebra of a Lie group, exponential map, examples of Lie algebra, covering G with one-parameter subgroups?, subgroups and subalgebras, commutators of matrices, left invariant vector fields generate right translations)

• Chapter 16: Vector Bundles in Geometry and Physics ( vector bundles, fiber coordinates, transition functions, local trivialization, normal bundle to submanifold, Poincare's Theorem and the Euler Characteristic, Hopf's Theorem, connections in a vector bundle, covariant derivative, curvature, complex vector space, structure group of bundle, complex line-bundles, The Electromagnetic Connection, Weyl's principle of Gauge invariance, global potentials, the Dirac Monopole, Aharonov-Bohm Effect)

• Chapter 17: Fiber Bundles, Gauss-Bonnet, and Topological Quantization (fiber bundles, principal bundles, frame bundles, action of structure group on principal bundle, coset spaces, transitive actions, free actions, stability, isotropy, little subgroup, homogeneous space, Grassmann manifolds, Chern's proof of the Gauss-Bonnet-Poincare-Theorem, Gauss-Bonnet as an Index Theorem, generalizations of Gauss-Bonnet, hermitian line-bundles, index, Chern forms, intersection number, topological quantization condition, Berry Phase, monopoles and the Hopf Bundle)

• Chapter 18: Connections and Associated Bundles (Maurer-Cartan form, Lie-algebra valued forms on a manifold, Maurer-Cartan equation, anticommutator, connections in a principal bundle, $$G$$-frames, horizontal distribution, principal bundle, representation, associated bundle through representation, connections in associated bundles, Adjoint bundle, sections of vector bundle, curvature of Ad bundle)

• Chapter 19: The Dirac Equation (the groups $$SO(3)$$ and $$SU(2)$$, rotation group, Lie algebra $$\mathfrak{su}(2)$$, Pauli matrices, $$SU(2)$$ is topologically the $$3$$-Sphere, adjoint map from $$SU(2)$$ to $$SO(3)$$ in detail, spinors and rotations of $$\mathbb{R}^3$$, Hamilton and quaternions, Clifford algebras, Dirac as squareroot of d'Alembertian, Lorentz group, $$SU(2)$$ is deformation retract of $$SL(2, \mathbb{C})$$ and $$SO(3)$$ is a deformation retract of $$L_0$$, Dirac algebra, Dirac Spinors, Dirac Operator, spinor bundle, spin connection)

• Chapter 20: Yang-Mills Fields (tensorial nature of Lagrange's equations, Noether's Theorem for internal symmetries, Noether Principal relating symmetries and conservation laws, Dirac Lagrangian, building scalars from spinors, Weyl's gauge invariance revisited, Electromagnetic Lagrangian, quantization of the field: photons, Heisenberg nucleon, Yang-Mills nucleon, field strength, quarks, gluons, charge, compact groups and Yang-Mills action, Yang-Mills equation, Yang-Mills analogy with electromagnetism, instantons, pure gauges, instanton winding number, instantons and the vacuum, tunneling and independent vacua )

• Chapter 21: Betti Numbers and Covering Spaces (bi-invariant forms, Cartan p-forms, bi-invariant Riemannian metrics, geodesics as one-parameter subgroups or their translates, harmonic forms in the bi-invariant metric, bi-invariant forms are harmonic w.r.t. the bi-invariant metric, Weyl's theorem on vanishing Betti number, Cartan's Theorem for the existence of a nontrivial harmonic 3-form, Poincare's fundamental group $$\pi_1(M)$$, homotopy of loops, simply connected, covering space, universal covering space, orientable covering, lifting paths, universal covering group, Theorem of S.B. Myers, connection of bi-invariant metric, Weyl's theorem about finite fundamental group)

• Chapter 22: Chern Forms and Homotopy Groups (Yang-Mills "winding number", winding number in terms of field strength, Chern-Simons 3-form, Chern Forms on $$U(n)$$ bundle, Theorem of Chern and Weil, homotopy, covering homotopy, topology of $$SU(n)$$, higher homotopy groups, homotopy groups of spheres, exact sequences of groups, boundary homomorphism, relation between homotopy and homology groups, Hurewicz theorem, some computations of homotopy groups, Hopf map, Hopf fibration, Chern forms as obstructions, Chern's integral)

My recommendations from the books I happen to have on hand:

1. Once again Topology, Geometry and Gauge Fields: Foundations and Topology, Geometry and Gauge Fields: Interactions by Gregory L. Naber. Also, Shigeyuki Morita's Geometry of Differential Forms has much to help with later chapters in Frenkel.
2. Mikio Nakahara's Geometry, Topology and Physics has a lot of discussion of monopoles and instantons, Yang-Mill's equations and more.
3. David Bleecker's Gauge Theory and Variational Principles builds the framework needed to write Yang-Mills equations in spacetime. Plenty of details on representations of the Lorentz group, connections on fiber bundles etc. This appears in the Bibliography of Frankel with good reason.
4. Stephen Bruce Sontz's Principal Bundles: The Classical Case this text hits a lot of the high points in Frankel. It is also unusual in its approach to using some category theory to streamline discussion. It builds enough to do Yang-Mills, I don't see much Topology or index theory, but it is intended as more of an introductory treatment. There is certainly an audience for this text. It would have been invaluable to me at one point in my history.
5. Lewis H. Ryder's Quantum Field Theory in particular Section 3.3 gives a window into how Physicist's came to demand the transformation rule for a potential one-form in electromagnetism. It helps to appreciate the derivation of that same transformation rule arising naturally from the geometry of principal fiber bundles. Of course there is also plenty on Lorentz groups, Pauli matrices and all that good stuff.

## Other Texts of Note:

Similar books which I also tend to glance in for topics in Frankel:

• Paul Renteln's Manifolds, Tensors, and Forms I think it is fair to say Renteln is a shorter more disciplined version of Frenkel. It has similar cross-disciplinary goals. Renteln has a lot of wisdom and experience in both math and physics and that text is a treasure.

• M. Gockeler and T. Schucker's Differential Geometry, Gauge Theories, and Gravity is a relatively short Cambridge monograph (230 pages) so you can surmise the detail. This has great sentimental value to me in that my advisor recommended it as a text to understand why Maxwell's Equations were consistent with Special Relativity. I read it, waiting to find that in there, well... I think its not there (I wanted what is in Resnik's text, gory partial derivative change of variable stuff), but it introduced me to this wild world of abstract bundles and beautiful geometry. No regrets.

• R.W.R. Darling's Differential Forms and Connections I also read this as an undergraduate and I think a lot of it slipped past me. However, I think this is where I first learned of the flux and work form maps which are essential for understanding the interplay between the differential forms and vectors on $$\mathbb{R}^3$$. I probably would learn a lot if I reread this book now.

• Bernard Schutz Geometrical Methods of Mathematical Physics on occasion I find formulas in this book are really helpful.

Well, I'm sure there are more to add. It occurs to me that Frankel's text may well be the Kevin Bacon for math physics books I treasure most.

• +1 for a fantastic reading list for graduate students in mathematics interested in physics as well as serious physics graduate students (my mentor Nick Metas always told me that when he was a student at MIT, you could always tell the serious physics graduate students because they took mathematics courses in the MATH department!) Apr 11 '20 at 0:26

Feeling bad for you with no answers.

It might (or might not!) be opposite to the direction you need, but you take a look at the Kreysig book in this area. He was a mathematician but with a very strong sympathy for engineers and physicists. And in any case, it is very strong pedagogically (progression of ideas based on student learning not math efficiency, answers to exercises in the back, etc.)

https://www.amazon.com/Differential-Geometry-Dover-Books-Mathematics/dp/0486667219