curve

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

An example: elliptic curve.

**Examples of sequences of local structures**

geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|

$\leftarrow$ differentiation | integration $\to$ | |||||||

smooth functions | derivative | Taylor series | germ | smooth function | ||||

curve (path) | tangent vector | jet | germ of curve | curve | ||||

smooth space | infinitesimal neighbourhood | formal neighbourhood | germ of a space | open neighbourhood | ||||

function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||

arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||

Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||

symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |

Revised on December 18, 2014 16:11:22
by Urs Schreiber
(127.0.0.1)