Narrow Results

In this paper, we derive a topological obstruction to the removal of an isolated degenerate complex tangent to an embedding of a 3-manifold into ℂ³ (without affecting the structure of the remaining complex tangents). We demonstrate how the vanishing of this obstruction is both a necessary and sufficient condition for the (local) removal of the isolated complex tangent. The obstruction is a certain homotopy class of the space 𝕐 consisting of totally real 3-planes in the Grassmannian of real 3-planes in ℂ³(= ℝ⁶). We further compute additional homotopy and homology groups for the space 𝕐 and of its complement 𝕎 consisting of "partially complex" 3-planes in ℂ³.

We show that, for a closed orientable $n$-manifold, with $n$ not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex $(n-1)$-space ensures the existence of a totally real embedding into complex $n$-space. This implies that a closed orientable $(4k+1)$-manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex $4k$-space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds.

In this paper, we give a characterization of homogeneous totally real minimal two-spheres in a complex hyperquadric $Q_n$. Let $f$ be a totally real minimal immersion from two-sphere in $Q_n$, and ${\tau }_{XY},{\tau }_{Xc}$ (see Sec. 2) are globally defined invariants relative to the first and second fundamental forms. We prove that if its Gauss curvature $K$ and ${\tau }_{XY}$ are constants, and ${\tau }_{Xc}$ vanishes identically, then $f$ is congruent to $F_{2k,2l}$ constructed by the Boruvka spheres with $n=2(k+l)$.