# 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) $0$.

## 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 infinitesimal and local structures

first order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ differentiationintegration $\to$
derivativeTaylor seriesgermsmooth function
tangent vectorjetgerm of curvecurve
square-0 ring extensionnilpotent ring extensionring extension
Lie algebraformal grouplocal Lie groupLie group
Poisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

Revised on April 5, 2014 03:19:08 by Urs Schreiber (185.37.147.12)