nLab curve



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)


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics


In differential geometry

For XX a smooth manifold, a (parametrized oriented) smooth curve in XX is a smooth function γ:X\gamma\colon \mathbb{R} \to X from the real line (or an interval therein) to XX. (Compare path.)

For most purposes in differential geometry one needs to work with a regular curve, which is a parametrized smooth curve whose velocity, i.e. the derivative with respect to the parameter, is never zero. For example, this is important if one wants to split curve into segments which have no self-intersections, which is important.

In the foundations of differential topology, it is possible to define a tangent vector as an equivalence class of smooth curves at a given point in the image of the curve, effectively identifying a curve with its derivative at (say) 00.

See also the fundamental theorem of differential geometry of curves?.

Open and closed curves

In a Cartesian space n\mathbb{R}^n, an open smooth curve is a smooth curve γ: n\gamma:\mathbb{R} \to \mathbb{R}^n which does not intersect itself: For every real number aa \in \mathbb{R} and bb \in \mathbb{R}, the distance between the two points on the curve parameterized by aa and bb is greater than zero: ρ(γ(a),γ(b))>0\rho(\gamma(a),\gamma(b)) \gt 0. Equivalently, an open smooth curve is a smooth curve such that the shape of the image of γ\gamma is contractible: ʃim(γ)*\esh \mathrm{im}(\gamma) \simeq *.

A closed smooth curve is a smooth curve γ: n\gamma:\mathbb{R} \to \mathbb{R}^n which does intersect itself: for every real number aa \in \mathbb{R}, there is a real number bb \in \mathbb{R}, such that γ(a)=γ(a+b)\gamma(a) = \gamma(a + b). Equivalently, a closed smooth curve is a smooth curve such that the shape of the image of γ\gamma is equivalent to the circle type: ʃim(γ)S 1\esh \mathrm{im}(\gamma) \simeq S^1.

In classical mathematics, every smooth curve γ: n\gamma:\mathbb{R} \to \mathbb{R}^n is either open or closed. In constructive mathematics, there are smooth curves where it cannot be proved to be either open or closed, due to the failure of trichotomy.

In algebraic geometry

In algebraic geometry, an algebraic curve is a 11-dimensional algebraic variety over a field.

An example: elliptic curve.


See also

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

Last revised on May 18, 2022 at 18:28:29. See the history of this page for a list of all contributions to it.