nLab differential geometry



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Higher geometry



Differential geometry is a mathematical discipline studying geometry of spaces using differential and integral calculus. Classical differential geometry studied submanifolds (curves, surfaces…) in Euclidean spaces. The traditional objects of differential geometry are finite and infinite-dimensional differentiable manifolds modelled locally on topological vector spaces. Techniques of differential calculus can be further stretched to generalized smooth spaces. One often distinguished analysis on manifolds from differential geometry: analysis on manifolds focuses on functions from a manifold to the ground field and their properties, together with applications like PDEs on manifolds. Differential geometry on the other hand studies objects embedded into the manifold like submanifolds, their relations and additional structures on manifolds like bundles, connections etc. while the topological aspects are studied in a younger branch (from 1950s on) which is called differential topology.

Generalized smooth spaces from nnPOV

See also generalized smooth space.

Finite-dimensional differential geometry is the geometry modeled on Cartesian spaces and smooth functions between them.

Formally, it is the geometry modeled on the pre-geometry 𝒢=\mathcal{G} = CartSp.

This includes a sequence of concepts of generalized smooth spaces:

Similarly, standard models of synthetic differential geometry in higher geometry are modeled on the pre-geometry 𝒢=\mathcal{G} = ThCartSp. To wit, the cohesive topos Sh(ThCartSp)Sh(ThCartSp) is the smooth topos called the Cahiers topos:


local modelglobal geometry
Klein geometryCartan geometry
Klein 2-geometryCartan 2-geometry
higher Klein geometryhigher Cartan geometry


geometries of physics

A\phantom{A}(higher) geometryA\phantom{A}A\phantom{A}siteA\phantom{A}A\phantom{A}sheaf toposA\phantom{A}A\phantom{A}∞-sheaf ∞-toposA\phantom{A}
A\phantom{A}discrete geometryA\phantom{A}A\phantom{A}PointA\phantom{A}A\phantom{A}SetA\phantom{A}A\phantom{A}Discrete∞GrpdA\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}CartSpA\phantom{A}A\phantom{A}SmoothSetA\phantom{A}A\phantom{A}Smooth∞GrpdA\phantom{A}
A\phantom{A}formal geometryA\phantom{A}A\phantom{A}FormalCartSpA\phantom{A}A\phantom{A}FormalSmoothSetA\phantom{A}A\phantom{A}FormalSmooth∞GrpdA\phantom{A}


See also references at Riemannian geometry.

Diff geometry of curves and surfaces

The study of differential geometry goes back to the special case of differential geometry of curves and surfaces:

the study of curves and surfaces embedded into Euclidean space 3\mathbb{R}^3:

General but traditional diff geometry

With emphasis on G-structures:

With emphasis on natural bundles:

With emphasis on Cartan geometry:

Lecture notes:

Introductions with an eye towards applications in (mathematical)physics, specifically to gravity and gauge theory:

A discussion in the context of Frölicher spaces and diffeological spaces:

See also

Higher diff geometry

See at higher differential geometry.

Derived diff geometry

For derived differential geometry see

Last revised on July 11, 2024 at 10:45:43. See the history of this page for a list of all contributions to it.