Concept map

Differentiable manifolds

A manifold is a space that looks like ℝ^m up close. The course makes “up close” precise, then carries calculus across it one tool at a time.

1. Manifolds (I). Charts glue Euclidean pieces together, and smoothness is read through them.
2. Linearisation (II–III). Tangent spaces and differentials linearise maps, and the rank of the differential controls the local shape of a map.
3. Dynamics and symmetry (IV–V). Vector fields integrate to flows, and Lie groups are manifolds that carry their own symmetries.
4. Forms and metrics (VI–VII). Forms can be integrated, which leads to Stokes and degree. A metric adds lengths, connections and geodesics.

Click or tap a box to open its summary. Hover over a box to highlight its prerequisites and consequences. Dependencies not drawn on the map are listed in each box’s detail panel.