Smooth projective translation planes
A projective or affine plane is called smooth if point and line space are endowed with smooth structures such that the geometric operations of joining and intersecting are smooth. The author of the paper under review has produced examples of smooth projective planes and of smooth affine translation planes. However, the latter examples do not extend to smooth projective planes, except if one has the plane over one of the real algebras \(\mathbb{R}, \mathbb{C}, \mathbb{H}\), or \(\mathbb{O}\). In the paper under review, it is shown that in fact these four planes are the only smooth projective translation planes. The proof starts with a characterization of those elements of \(\text{GL}(n,\mathbb{R})\) that extend to diffeomorphisms of the sphere \(\mathbb{R}^n \cup \{\infty\}\), with its natural smooth structure. This characterization is used to show that the group of projectivities of a line in a smooth projective translation plane is comparatively small; a result of Grundhöfer and Strambach then yields the theorem.
scientific article; zbMATH DE number 826781
scientific article