Eversion

A curve on the unit sphere $\Bbb{S}^2$ is an eversion if it has no corners or cusps (but it may be self-intersecting). These properties are guaranteed by requiring that the curve's velocity never vanishes. A mapping $\boldsymbol{\sigma}:\Bbb{S}^1\to \Bbb{S}^2$ forms an immersion of the Circle into the Sphere Iff, for all $\theta\in\Bbb{R}$,

$$\left\vert{{d\over d\theta} [\boldsymbol{\sigma}(e^{i\theta})]}\right\vert>0.$$

Smale (1958) showed it is possible to turn a Sphere inside out (Sphere Eversion) using eversion.

See also Sphere Eversion

References

Smale, S. ``A Classification of Immersions of the Two-Sphere.'' Trans. Amer. Math. Soc. 90, 281-290, 1958.