 

Ali M. Elgindi
On the topological structure of complex tangencies to embeddings of S^{3} into C^{3} view print


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


Abstract
In the mid1980's, M. Gromov used his machinery of the hprinciple to prove that there exists totally real embeddings of S^{3}
into C^{3}. 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 codimension2 subspaces. We first consider the Heisenberg group H and generate some interesting results therein. Then, by using the biholomorphism of H with the 3sphere minus a point, we
demonstrate that every homeomorphismtype of knot in S^{3} may arise precisely as the set of complex tangents to an embedding S^{3} ⟶ C^{3}. 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
alielgindi@gmail.com

