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:
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,
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)
