For a smooth manifold, a (parametrized oriented) smooth curve in is a smooth function from the real line (or an interval therein) to . (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) .
See also the fundamental theorem of differential geometry of curves?.
In algebraic geometry, an algebraic curve is a -dimensional algebraic variety over a field.
An example: elliptic curve.
Examples of sequences of local structures
Last revised on February 12, 2021 at 06:33:36. See the history of this page for a list of all contributions to it.