nLab curve

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) $0$.

See also the fundamental theorem of differential geometry of curves?.

Open and closed curves

In a Cartesian space $\mathbb{R}^n$, an open smooth curve is a smooth curve $\gamma:\mathbb{R} \to \mathbb{R}^n$ which does not intersect itself: For every real number $a \in \mathbb{R}$ and $b \in \mathbb{R}$, the distance between the two points on the curve parameterized by $a$ and $b$ is greater than zero: $\rho(\gamma(a),\gamma(b)) \gt 0$. Equivalently, an open smooth curve is a smooth curve such that the shape of the image of $\gamma$ is contractible: $\esh \mathrm{im}(\gamma) \simeq *$.

A closed smooth curve is a smooth curve $\gamma:\mathbb{R} \to \mathbb{R}^n$ which does intersect itself: for every real number $a \in \mathbb{R}$, there is a real number $b \in \mathbb{R}$, such that $\gamma(a) = \gamma(a + b)$. Equivalently, a closed smooth curve is a smooth curve such that the shape of the image of $\gamma$ is equivalent to the circle type: $\esh \mathrm{im}(\gamma) \simeq S^1$.

In classical mathematics, every smooth curve $\gamma:\mathbb{R} \to \mathbb{R}^n$ is either open or closed. In constructive mathematics, there are smooth curves where it cannot be proved to be either open or closed, due to the failure of trichotomy.

In algebraic geometry

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

An example: elliptic curve.

Related concepts

• surface

• differential geometry of curves and surfaces

• singular point of a curve

• torsion of a curve

• surface, hypersurface

• velocity, acceleration, jerk

References

See also

• Wikipedia, 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