Differential Geometry & Fiber Bundles

Reference notes on global differential geometry — the geometry of bundles over a manifold and the topology of fields — as opposed to the metric geometry of a single space (which lives in geometry/). This is the mathematical backbone of gauge theory: a gauge field is a connection on a principal bundle, its field strength is the curvature, a Wilson loop is holonomy, a topological charge is a characteristic number, and an anomaly is an index.

These pages are companion material to the Mathematics section and the Physics tree. They start from the smooth-manifold and exterior-calculus machinery already developed in topology-manifolds.md and analysis/differential-forms.md, and supply the structure that QFT gauge theory, anomalies, and topological field configurations take for granted.

Scope vs. the geometry folder

geometry/ owns the metric story — Euclid's postulates, hyperbolic/elliptic geometry, and Riemannian curvature of a space. This folder owns the bundle story — connections and curvature on bundles over a space, their global invariants, and the homotopy classification of field configurations. The two meet at exactly one point: the Levi-Civita connection is the special case "connection on the tangent bundle", spelled out in riemannian-bridge.md.

Contents

A. Manifold calculus

1. Vector Fields, Flows, and the Lie Derivative — sections of $TM$, integral curves, the Lie bracket and Lie derivative, and the Frobenius theorem.
2. Tensor Fields and Differential Forms — tensor bundles, $p$-forms, pullback, the exterior derivative, and integration on manifolds.
3. De Rham Cohomology — closed vs. exact forms, the Poincaré lemma, and the de Rham theorem tying calculus to topology.

B. Fiber bundles

4. Fiber and Vector Bundles — local trivializations, transition functions, sections, and the frame bundle.
5. Principal Bundles — the structure group, gauge transformations, and the classification of bundles.
6. Associated Bundles and Matter Fields — the associated vector bundle, the adjoint bundle, and structure-group reduction.

C. Connections and curvature

7. Connections and Parallel Transport — the connection 1-form, the covariant derivative $\nabla =d+A$, and gauge transformations.
8. Curvature — the curvature 2-form, the structure equation, the Bianchi identity, and the dictionary $\Omega \leftrightarrow F_{\mu \nu }$.
9. Holonomy and Wilson Loops — holonomy, the Ambrose–Singer theorem, path-ordered exponentials, and Aharonov–Bohm.

D. Characteristic classes and the index theorem

10. Chern–Weil Theory — invariant polynomials of the curvature as topological invariants.
11. Characteristic Classes — Chern, Pontryagin, and Euler classes; the instanton number and the $\theta$-term.
12. The Atiyah–Singer Index Theorem — analytical index = topological index, and the chiral anomaly as $\mathrm{index}(/D)$.

E. Topology of field configurations

13. Homotopy Groups and the Classification of Defects — ${\pi }_n$, the homotopy groups of Lie groups and spheres, and topological charge.
14. Defects, Solitons, Instantons, and Monopoles — the homotopy classification of stable field configurations.

F. Bridges

15. Riemannian Geometry as a Tangent-Bundle Connection — the Levi-Civita connection as the metric instance of the general theory.
16. Why This Matters — The Gauge-Theory Bridge — the full geometry ↔ gauge-theory dictionary.

Reading order

The spine runs $$\text{vector-fields}\to \text{tensors-and-forms}\to \text{de-rham}\to \text{fiber-bundles}\to \text{principal-bundles}\to \text{connections}\to \text{curvature},$$ after which holonomy, Chern–Weil, and characteristic classes build the global invariants, and homotopy + defects classify field configurations. The index theorem and the two bridge pages sit on top. Manifold and exterior-calculus prerequisites come from topology-manifolds.md §2–3 and analysis/differential-forms.md; the structure groups are the Lie groups of group-theory/. Readers who only need a specific definition can jump in anywhere.