Differential Geometry – Smooth Mapping from $\mathrm{RP}^2$ to $\mathbb{R}^3$ with Nonsingular Derivative

dg.differential-geometrydifferential-topology

I'm trying to find a mapping $f$ from the 2D real projective plane to $\mathbb{R}^3$ which

  1. is smooth

  2. has non-singular directional derivative. That is,

    $\forall x, v, \quad v \ne 0 \implies D_v f(x) \ne 0$.

This is different from an embedding, which is impossible because the surface would have to intersect itself. Here, I am allowing this intersection.

1 and 2 are possible for at least one other non-orientable manifold—the Klein bottle. Is there some simple way of bending the real projective plane such that it is smooth and has non-singular derivative in $\mathbb{R}^3$, like the Klein bottle?

Best Answer

This is a famous problem that was solved by a doctoral student of David Hilbert named Werner Boy in 1901. The kind of mapping you are looking for is called an "immersion" (of its domain — the real projective plane — into 3-space).

The surface Boy discovered is now called "Boy's surface" and there are plenty of references if you search for that name. Perhaps surprisingly, it has threefold symmetry. (There are many immersions of the projective plane in 3-space, but Boy's is surely the simplest.) It is known that all such immersions must contain a triple point: a point in 3-space whose inverse image by the immersion is of cardinality at least 3.

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