Rigidity of the canonical isometric imbedding of the Cayley projective plane \(P^2(\mathrm{Cay})\) (Q2574436)

In a former paper [Hokkaido Math. J. 33, 399--412 (2004; Zbl 1094.53055)], the authors proved that \(P^2\)(Cay) cannot be isometrically immersed into \(\mathbb R^{25}\) even locally. In the present paper, the authors investigate isometric immersions of \(P^2\)(Cay) into \(\mathbb R^{26}\) and prove that the canonical isometric imbedding \(f_0\) of \(P^2\)(Cay) into \(\mathbb R^{26}\), which was defined by \textit{S. Kobayashi} [Tohoku Math. J. 20, 21--25 (1968; Zbl 0175.48301)], is rigid in the following strongest sense: Any isometric immersion \(f_1\) of a connected open set \(U(\subset P^2(\text{Cay}))\) into \(\mathbb R^{26}\) coincides with \(f_0\) up to a Euclidean transformation of \(\mathbb R^{26}\), i.e., there is a Euclidean transformation \(a\) of \(\mathbb R^{26}\) satisfying \(f_1=af_0\) on \(U\).