Alphabeta MathDifferential Geometry
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.
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
2 pagesReal analysis already supplies local existence, uniqueness, continuous dependence, and maximal continuation for Euclidean first-order ODEs.
2 definitions, 1 lemma, 3 propositions, 5 theorems, 2 corollaries, 4 false statementsExamples & counterexamples →- Smooth Manifolds and Smooth Maps40 results
This page separates the topological manifold axioms from the atlas and maximal atlas that define a smooth structure, then proves that smoothness of a map does not depend on the chosen smooth charts.
9 definitions, 6 lemmas, 15 propositions, 3 theorems, 6 false statements, 1 remarkExamples & counterexamples →
Level 1
2 pagesThis page builds smooth partitions of unity from explicit flat and step functions, Euclidean and manifold bumps, and locally finite normalization.
4 definitions, 11 lemmas, 1 proposition, 6 theorems, 4 corollaries, 6 false statements, 1 remarkExamples & counterexamples →- Tangent Cotangent and the Differential49 results
This page defines tangent vectors intrinsically as derivations on smooth germs, identifies them with curve velocities, derives the differential and its coordinate formula…
13 definitions, 11 lemmas, 4 propositions, 12 theorems, 3 corollaries, 6 false statementsExamples & counterexamples →