Alphabeta Math

Differential Geometry

60 pages in 1 part

Differential geometry studies spaces that are locally modelled on Euclidean space while retaining the global structure that coordinates can obscure. The collection begins with smooth manifolds and maps, then develops smooth bump functions, partitions of unity and exhaustions—the tools that turn local constructions into global ones. Tangent and cotangent spaces make first-order behaviour intrinsic, and the differential records how a smooth map acts on that behaviour. Euclidean ordinary differential equations supply the local existence, uniqueness and smooth-dependence results from which vector fields and flows are built.

Later pages extend this foundation through rank theorems and embedded submanifolds, vector bundles, transversality, differential forms, Stokes' theorem, de Rham theory, orientations, Riemannian geometry, curvature and geodesics. The category relies on topology for manifold separation and countability hypotheses, and on real analysis for multivariable differentiation, inverse and implicit function theorems, integration and change of variables. Algebraic topology and geometric analysis then reuse its intrinsic language rather than rebuilding it inside their own tracks.

Lie groups, homogeneous spaces, Lie-algebra structure theory and their representations now belong to Lie Theory. That category uses these general manifold foundations and keeps the corresponding prerequisites explicit.

Pathway

The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.

  1. Part 1 · Smooth Manifolds and Geometric Structures

    30 pages

    Smooth manifolds, tangent spaces, submanifolds, vector bundles and flows open the pathway. Differential forms lead to Stokes and de Rham theory, while metrics and connections lead to geodesics, curvature, Jacobi fields and comparison, and Gauss–Bonnet. On compact Hermitian vector bundles, Laplace-type operators with positive principal symbol admit parametrices and a Fredholm Hodge decomposition whose finite smooth kernel gives harmonic representatives of de Rham and Dolbeault cohomology. Symplectic geometry and characteristic forms complete the core; Lie groups and Lie-algebra structure theory have their own category, Lie Theory.