The homeomorphism type of unital-like submanifolds in smooth projective planes
A projective plane such that the point space \(P\) and the line space \(\mathcal L\) are manifolds and joining and intersecting are smooth operations is said to be a smooth plane. \(P\) and \(\mathcal L\) are then homeomorphic to the point space of a classical plane \({\mathcal P}_2{\mathbb F}\) where \(\mathbb F\) is the division algebra \(\mathbb O\) of the real octonions or a subfield \(\mathbb R\), \(\mathbb C\), or \(\mathbb H\) of \(\mathbb O\). In particular, \(\dim{\mathbb F} = \ell \mid 8\) and \(\dim P = 2\ell\). The set \(U_\pi \) of absolute points of a smooth polarity \(\pi\) in a smooth plane is called a smooth polar unital if it is not empty. Theorem: If \(U\) is a smooth polar unital, then \(U\) is homeomorphic to a sphere \({\mathbb S}_{2\ell-1}\) or \({\mathbb S}_{{3\over2}\ell-1}\), that is, the unital of a hyperbolic polarity in a classical plane, an orthogonal polarity in \({\mathcal P}_2{\mathbb C}\), or a planar polarity in the quaternion or octonion plane. More generally, the following is true: (\(*\)) If \(U\) is a smoothly embedded submanifold of the point space of a smooth plane, if there is precisely one tangent at each point of \(U\), and if each secant intersects \(U\) transversally, then \(U\) is homeomorphic to a sphere. Assertion (\(*\)) is proved by means of Ehresmann's fibration theorem. The theorem is then a consequence of (\(*\)) and results by \textit{S. Immervoll} [Result. Math. 39, No. 3--4, 218--229 (2001; Zbl 1017.51014)].
scientific article; zbMATH DE number 2051656