In general, a curve refers to a differentiable Continuous path from an interval of the reals to a Differentiable manifold

However, if the codomain manifold is not specified, one often assumes it refers to a spatial curve, that is a curve in R^3 (that is codomain is $\mathbb{R}^3$)

Definition of regular curve, as opposed to singular ones, which may have Cusps and nodes. They have arc length parametrization (p.a.l). They have no intrinsic geometry

A curve is a differentiable Continuous path from an interval of the reals to R^3, $x(t)$. The curve is regular if $x'(t) \neq 0$ for all $t$.

basis (Frenet trihedron). The derivative of the Tangent vector $T$ is orthogonal to the tangent vector. The norm of this vector $k=|T'|$ is the curvature of the curve at the point. The normal vector is the normalized version of this vector $N=T'/k$. To complete the basis we introduce the binormal vector $B=T \times N$. The magnitude of the derivative of the binormial vector defines the torsion. One can then derive Frenet formulae

The Fundamental theorem of the local theory of curves states that given two smooth functions, $k(s) > 0$ and $\tau(s)$, there exists a curve $\alpha(s)$ p.a.l., such that its curvature is $k$ and its torsion is $\tau$. Moreover, $\alpha$ is unique (up to direct isometries (rigid motions of $R^3$, with determinant $1$))

Curvature of a curve

Curve on a surface

Interactive curves on manifolds:

https://www.dgpad.net/index.php?url=http://curvica974.re/EnLigne/M1Maths/SolenoideTore.dgp

https://www.dgpad.net/index.php?url=http://curvica974.re/EnLigne/M1Maths/GeodesiqueDuToreAvecT.dgp

https://www.dgpad.net/index.php?url=http://curvica974.re/EnLigne/M1Maths/SphereOsculatrice.dgp