Curves

# 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) $0$.

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

## In algebraic geometry

In algebraic geometry, an algebraic curve is a $1$-dimensional algebraic variety over a field.

An example: elliptic curve.

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 February 12, 2021 at 06:33:36. See the history of this page for a list of all contributions to it.