Toward differential and symplectic geometry

Roll a sheet of paper into a cylinder without stretching it. Distances measured along the sheet are unchanged locally, even though its appearance in three-dimensional space has changed. A sphere behaves differently: flattening a patch onto a plane cannot preserve all lengths. This is an inviting first question in differential geometry: which features belong to the geometry experienced on a surface, and which depend on how that surface sits in another space? Shifrin's treatment of the first fundamental form and Gauss's theorem makes that distinction calculable.

The route soon extends beyond surfaces. A configuration space for a mechanism, a family of subspaces and a space of positions and momenta may all need local coordinates, without having one convenient global coordinate system. Smooth manifolds make calculus possible in that setting. The aim is to recognize a tangent vector, a differential form or a geometric structure through descriptions that agree when coordinates change. This is why coordinate calculations and abstract definitions should develop together.

From coordinates to manifolds

Start with Nigel Hitchin's Differentiable Manifolds, the 2014 Oxford course notes. Sections 2–4 develop manifolds, tangent vectors and vector fields; §§5–7 lead through exterior algebra, differential forms and integration. Work with the sphere and torus as recurring examples. An early goal is to describe a tangent vector in two charts and check that the descriptions agree. Later, compute a pullback of a differential form and explain the role of orientation in an integral.

The preparation is concrete: linear maps and dual spaces, the multivariable chain rule, inverse and implicit function theorems, and basic topology. You should be comfortable distinguishing a local construction from a global statement. If you can construct charts on a sphere and differentiate maps between coordinate patches, skip a general calculus review. If notation for tangent spaces is the main obstacle, return to derivatives as linear maps. Knowing all of algebraic topology in advance is unnecessary; add global invariants when examples give them a purpose.

Two other entry points

One alternative is Theodore Shifrin's Differential Geometry: A First Course in Curves and Surfaces. Choose it when you want curvature and geodesics to emerge from explicit calculations before moving to abstract manifolds. Chapter 2 develops surface geometry through fundamental forms, the Gauss map and covariant differentiation. Compare a plane, cylinder and sphere, calculating enough to see which curvature quantities detect bending and which detect intrinsic geometry.

For readers who already know manifolds and forms, the other alternative is Ana Cannas da Silva's Lectures on Symplectic Geometry. Begin with Lectures 1–2 on symplectic forms and cotangent bundles. Here the extra structure is a closed, nondegenerate two-form; it does not by itself specify lengths and angles. Lecture 18 connects that structure with Hamiltonian dynamics. Positions and momenta provide a useful running example, especially when coordinate changes make the familiar equations look different.

Metrics are not symplectic forms

A common misconception is that symplectic geometry is Riemannian geometry with a different matrix representing the metric. Darboux's theorem, treated in Lecture 8, shows that symplectic forms of a fixed dimension have a common local normal form. Riemannian curvature has no direct counterpart of that kind. The interesting questions therefore organize themselves differently, with global phenomena playing a major role.

Choose the next branch through a question: shortest paths and curvature lead toward Riemannian geometry; Hamiltonian flows and Lagrangian submanifolds toward symplectic geometry; closed forms and integration toward de Rham cohomology. Connections with complex geometry or geometric PDE require additional tools selected for that destination. Before advancing, make sure you can name the structure a map preserves and test that condition in an example.

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