Rigidity of the canonical isometric imbedding of the Cayley projective plane $P^2(Cay)$

Abstract

In [7], we have proved that $P^2({\mathbf Cay})$ cannot be isometrically immersed into ${\mathbf R}^{25}$ even locally. In this paper, we investigate isometric immersions of $P^2({\mathbf Cay})$ into ${\mathbf R}^{26}$ and prove that the canonical isometric imbedding ${\mathbf f}_0$ of $P^2({\mathbf Cay})$ into ${\mathbf R}^{26}$, which is defined in Kobayashi~\cite{kobayashi}, is rigid in the following strongest sense: Any isometric immersion ${\mathbf f}_1$ of a connected open set $U (\subset P^2({\mathbf Cay}))$ into ${\mathbf R}^{26}$ coincides with ${\mathbf f}_0$ up to a euclidean transformation of ${\mathbf R}^{26}$, i.e., there is a euclidean transformation $a$ of ${\mathbf R}^{26}$ satisfying ${\mathbf f}_1=a{\mathbf f}_0$ on $U$.