 
Summary: THE HULL OF RUDIN'S KLEIN BOTTLE
JOHN T. ANDERSON
Abstract. In 1981 Walter Rudin exhibited a totally real embedding of the
Klein bottle into C2. We show that the polynomially convex hull of Rudin's
Klein bottle contains an open subset of C2. We also describe another totally
real Klein bottle in C2 whose hull has topological dimension equal to three.
1. Introduction
A real submanifold M of complex Euclidean space Cn
is said to be totally real if, at
each point p M, the tangent space Tp(M) to M (with the usual identifications of
Tp(M) with a subspace of R2n
) contains no nontrivial complex subspace. Among
the real surfaces, only the torus and connected sums of an odd number of Klein
bottles admit totally real embeddings into C2
(see for example [5]). The first explicit
example of a totally real embedding of the Klein bottle in C2
was given by Walter
Rudin in [7]. Section 20 of [3] describes a family of immersions of the Klein bottle
into C2
that includes Rudin's embedding.
