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

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.