Local isometric imbeddings of \(P^2(\mathbb H)\) and \(P^2(\mathbf{Cay})\) (Q1772572)

The authors study local isometric imbeddings of the quaternion projective plane \(P^2(\mathbb H)\) and the Cayley projective plane \(P^2(\mathbf{Cay})\) into Euclidean spaces. They prove: Let \(G/K\) be the quaternion projective plane \(P^2(\mathbb H)\) or the Cayley projective plane \(P^2(\mathbf{Cay})\). Then no open set of \(G/K\) can be isometrically imbedded into the Euclidean space \(\mathbb R^{q(G/K)}\). Accordingly, \(\mathbb R^{q(G/K)+1}\) is the least dimensional Euclidean space into which \(G/K\) can be locally isometrically imbedded. By this theorem, the authors conclude that the isometric imbeddings given by \textit{S. Kobayashi} [Tôhoku Math. J. (2) 20, 21--25 (1968; Zbl 0175.48301)], are the least dimensional isometric imbeddings of \(P^2(\mathbb H)\) and \(P^2(\mathbf{Cay})\).