Self-Intersection

A self-intersection of a parametrized curve or surface is an image point corresponding to two or more distinct points of its parameter domain. For a map , this means that for some . Self-intersections of smooth manifolds are studied through immersions (Whitney 1944; Lee 2013, pp. 77-97).

An immersion has an injective tangent map at every point, but need not be globally one-to-one. For example, the circle parametrized by modulo has an immersion into given by

The two distinct parameter values and have the same image , while the derivative never vanishes. Thus a self-intersection does not require a failure of local regularity. An embedding, in contrast, is one-to-one and has no self-intersections.

The Wente torus is an example of a self-intersecting immersed surface. Its self-intersections distinguish it from an embedded surface.