Complex Tangencies to Embeddings of Heisenberg Groups and Odd-Dimensional Spheres

TL;DR

This paper investigates the topological structure of complex tangents in embeddings of odd-dimensional spheres into complex spaces, extending previous results from 3D spheres and Heisenberg groups to higher dimensions.

Contribution

It generalizes earlier findings on complex tangents from 3D spheres and Heisenberg groups to higher odd-dimensional spheres and their embeddings.

Findings

Topological configurations of complex tangents are characterized for higher-dimensional spheres.

Results extend previous work to odd-dimensional spheres and their embeddings.

The structure of complex tangent sets is shown to be stratified manifolds of dimension (n-2).

Abstract

The notion of a complex tangent arises for embeddings of real manifolds into complex spaces. It is of particular interest when studying embeddings of real $n$-dimensional manifolds into $\mathbb{C}^n$. The generic topological structure of the set complex tangents to such embeddings $M^n \hookrightarrow \mathbb{C}^n$ takes the form of a (stratified) $(n-2)$-dimensional submanifiold of $M^n$. In this paper, we generalize our results from our previous work for the 3-dimensional sphere and the Heisenberg group to obtain results regarding the possible topological configurations of the sets of complex tangents to embeddings of odd-dimensional spheres $S^{2n-1} \hookrightarrow \mathbb{C}^{2n-1}$ by first considering the situation for the higher dimensional analogues of the Heisenberg group.