The Lens space is the quotient of by an action of . The group acts on by the root of unity in the first coordinate and the power of that in the second coordinate, and the action restricts to . So why a "Lens" space, and how does one visualize it?

 I drew by drawing a lattice in invariant under the group action. Here's the picture (after stereographic projection):

The map   restricted to is called the Hopf map and is usually viewed as a map (fibration) whose fibers (preimage of a point) are circles.

The preimage of a circle in is called a Clifford torus. These fill out all of except for two linked circles, and are in turn foliated by circles.

To visualize the Hopf fibration, we apply stereographic projection or, better yet, into the Heisenberg group. Here's a picture of three Clifford tori with Heisenberg horisontal lines:

Given a surface in Heisenberg space , the tangent space to at a given point is generally not the same as the horisontal plane. In fact, the set of points where the two are equal, called the critical points, at worst form a collection of lines on the surface.

 At every non-critical point , there is a single line in that is horisontal, and one may ask for a unit-speed horisontal line through . Given a parametrization of the surface, this becomes a system of PDE's that Mathematica can solve using NDSolve.

 In the case of the Euclidean unit sphere at the origin, there are two critical points: the north and south pole. The rest of the sphere is foliated by horisontal lines: