Module 01 — Smooth Manifolds Refresher
Why this module: A fiber bundle is a manifold built out of manifolds. Everything downstream — sections, connections, curvature — lives on manifolds, so the vocabulary must be automatic. You've seen this before (1.5 yrs ago); the goal is reactivation, not first contact.
Core idea
A Smooth Manifold is a space that locally looks like $\mathbb{R}^n$, with smooth transition between overlapping coordinate charts. The chart-overlap machinery here is a dry run for bundles: a bundle is defined the same way, with local trivializations playing the role of charts and transition functions gluing them.
Physics anchor: Minkowski spacetime is a manifold so simple (one global chart) that Landau never needs the general machinery. That's exactly why gauge theory looks non-geometric in Landau — the base is trivial, so all the geometry hides in the bundle over it.
Concepts to (re)work
- Smooth Manifold — charts, atlases, smooth maps
- Tangent Space — three equivalent definitions; pick your favorite
- Tangent Bundle — your first fiber bundle, before the general definition
- Vector Field — a section of the tangent bundle, before you know the word "section"
- Pushforward and Pullback — how maps move vectors and covectors
Reading
- Sternberg - Lectures on Differential Geometry, Ch. II (Differentiable Manifolds)
- Skim Landau §1–§7 to hold the physics in mind
Self-check before moving on
Can you state, without looking: what a chart is, what makes an atlas smooth, why a tangent vector at $p$ is not naturally an element of $\mathbb{R}^n$, and what data specifies a vector field?
Concepts in this module
Study tasks
- Read the five concept notes above and fill in their "My Notes" sections
- Sternberg Ch. II: read definitions of manifold, smooth map, tangent vector
- Exercise: write out two charts on $S^2$ (stereographic projections) and compute the transition function on the overlap
- Exercise: convince yourself $TS^2 \neq S^2 \times \mathbb{R}^2$ (hairy ball theorem) — this is *the* reason bundles are needed