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Ali M. Elgindi
On the topological structure of complex tangencies to embeddings of S3 into C3 view print
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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 |
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Abstract
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 S3 ⟶ C3. We also make note of the (nongeneric) situation where complex tangents arise along surfaces.
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Author information
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
alielgindi@gmail.com
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