On almost complex surfaces of the nearly Kaehler 6-sphere. II
On a 6-dimensional unit sphere \(S^ 6(1)\) a nearly Kaehler structure J can be defined in a natural way, making use of the Cayley numbers. In these two papers we consider almost complex surfaces M in \(S^ 6(1)\), i.e. surfaces whose tangent bundle is invariant under J. In [Kodai Math. J. 6, 174-185 (1983; Zbl 0517.53055)], \textit{K. Sekigawa} shows that, if an almost complex surface M has constant curvature K, the \(K=0\), \(K=1/6\), or \(K=1\). The problem of U. Simon is given by: ``Let M be an almost complex compact surface of \(S^ 6(1)\) with Gaussian curvature K. If 1/6\(\leq K\leq 1\), then \(K=1/6\) or \(K=1?\) If \(0\leq K\leq 1/6\), then \(K=0\) or \(K=1/6?''\) In these papers we give an affirmative answer to both questions. Our main tool is the integral formula of A. Ros. The results have been announced in Bull. Am. Math. Soc., New Ser. 19, 433-438 (1988; Zbl 0655.53052).
scientific article; zbMATH DE number 4109567