differential geometry

synthetic differential geometry

smooth topos

infinitesimal space

Kock-Lawvere axiom

integration axiom

microlinear space

synthetic differential supergeometry

. infinitesimal cohesion

de Rham space

formally smooth morphism, formally unramified morphism, formally étale morphism

jet bundle

Models for Smooth Infinitesimal Analysis

Fermat theory

smooth algebra ($C^\infty$-ring)

smooth locus

smooth manifold, formal smooth manifold, derived smooth manifold

smooth space, diffeological space, Frölicher space

smooth natural numbers

Cahiers topos

smooth ∞-groupoid

synthetic differential ∞-groupoid

tangent bundle,

vector field, tangent Lie algebroid;

differentiation, chain rule

differential forms

differential equation, variational calculus

connection on a bundle, connection on an ∞-bundle

Riemannian manifold

Hadamard lemma

Borel's theorem

Boman's theorem

Whitney extension theorem

Steenrod-Wockel approximation theorem

Poincare lemma

Stokes theorem

de Rham theorem

Chern-Weil theory

Lie theory, ∞-Lie theory

Chern-Weil theory, ∞-Chern-Weil theory

gauge theory

∞-Chern-Simons theory

Klein geometry, Klein 2-geometry, higher Klein geometry

Euclidean geometry, Cartan geometry, higher Cartan geometry

Riemannian geometry

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manifolds and cobordisms

manifold

coordinate chart, atlas

orientation, spin structure, string structure, fivebrane structure

smooth manifold, Riemannian manifold, complex manifold

cobordism

B-bordism

extended cobordism

cobordism category

(∞,n)-category of cobordisms

Thom spectrum

cobordism ring

genus

infinite dimensional manifold

signature genus, Kervaire invariant

A-hat genus, Witten genus

Thom's transversality theorem

Pontrjagin-Thom construction

Galatius-Tillmann-Madsen-Weiss theorem

geometrization conjecture,

Poincaré conjecture

elliptization conjecture

cobordism hypothesis-theorem

A manifold locally modeled on an infinite-dimensional Hilbert space.

Banach manifold,

Frechet manifold

convenient manifold

ILH manifold