New York Journal of Mathematics
Volume 18 (2012) 295-313


Ali M. Elgindi

On the topological structure of complex tangencies to embeddings of S3 into C3

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Published: May 9, 2012
Keywords: Complex tangents, real submanifolds of complex space, complex differential geometry, complex differential topology, 3-sphere, Heisenberg group
Subject: 32V40, 32V05, 57M25, 53C56

In the mid-1980's, M. Gromov used his machinery of the h-principle to prove that there exists totally real embeddings of S3 into C3. Subsequently, Patrick Ahern and Walter Rudin explicitly demonstrated such a totally real embedding. In this paper, we consider the generic situation for such embeddings, namely where complex tangents arise as codimension-2 subspaces. We first consider the Heisenberg group H and generate some interesting results therein. Then, by using the biholomorphism of H with the 3-sphere minus a point, we demonstrate that every homeomorphism-type of knot in S3 may arise precisely as the set of complex tangents to an embedding S3C3. We also make note of the (nongeneric) situation where complex tangents arise along surfaces.

Author information

Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia