curve

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 infinitesimal and local structures**

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

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

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

tangent vector | jet | germ of curve | curve | |||

square-0 ring extension | nilpotent ring extension | ring extension | ||||

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

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

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