Curve - Differential Geometry

Differential Geometry

While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general relativity, a world line is a curve in spacetime.

If is a differentiable manifold, then we can define the notion of differentiable curve in . This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take to be Euclidean space. On the other hand it is useful to be more general, in that (for example) it is possible to define the tangent vectors to by means of this notion of curve.

If is a smooth manifold, a smooth curve in is a smooth map

This is a basic notion. There are less and more restricted ideas, too. If is a manifold (i.e., a manifold whose charts are times continuously differentiable), then a curve in is such a curve which is only assumed to be (i.e. times continuously differentiable). If is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and is an analytic map, then is said to be an analytic curve.

A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two differentiable curves

and

are said to be equivalent if there is a bijective map

such that the inverse map

is also, and

for all . The map is called a reparametrisation of ; and this makes an equivalence relation on the set of all differentiable curves in . A arc is an equivalence class of curves under the relation of reparametrisation.