# nLab curve

Curves

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Curves

## In differential geometry

For $X$ a smooth manifold, a (parametrized oriented) smooth curve in $X$ is a smooth function $\gamma\colon \mathbb{R} \to X$ from the real line (or an interval therein) to $X$. (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) $0$.

### Open and closed curves

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

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

In classical mathematics, every smooth curve $\gamma:\mathbb{R} \to \mathbb{R}^n$ 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 $1$-dimensional algebraic variety over a field.

An example: elliptic curve.

## References

Examples of sequences of local structures

geometrypointfirst order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ 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 geometry$\mathbb{F}_p$ finite field$\mathbb{Z}_p$ p-adic integers$\mathbb{Z}_{(p)}$ 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.