On totally real spheres in complex space

Abstract

We shall prove that there are totally real and real analytic embeddings of $S^k$ in $\cc^n$ which are not biholomorphically equivalent if $k\geq 5$ and $n=k+2[\frac{k-1}{4}]$. We also show that a smooth manifold $M$ admits a totally real immersion in $\cc^n$ with a trivial complex normal bundle if and only if the complexified tangent bundle of $M$ is trivial. The latter is proved by applying Gromov's weak homotopy equivalence principle for totally real immersions to Hirsch's transversal fields theory.