ON TOTALLY REAL SPHERES IN COMPLEX SPACE

XIANGHONG GONG

Abstract. We shall prove that there are totally real and real analytic embeddings of S^{k} in C^{n} which are not biholomorphically equivalent if k ≥ 5 and n = k+2[\frac{k−1}{4}]. We also show that a smooth manifold M admits a totally real immersion in C^{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.