Hokkaido Mathematical Journal

AGAOKA Yoshio, KANEDA Eiji,

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

Hokkaido Mathematical Journal, 34 (2005) pp.331-353

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Abstract

In \cite{ak6}, we have proved that $P^2(\pmb{Cay})$ cannot be isometrically immersed into $\pmb{R}^{25}$ even locally. In this paper, we investigate isometric immersions of $P^2(\pmb{Cay})$ into $\pmb{R}^{26}$ and prove that the canonical isometric imbedding $\pmb{f}_0$ of $P^2(\pmb{Cay})$ into $\pmb{R}^{26}$, which is defined in Kobayashi~\cite{kobayashi}, is rigid in the following strongest sense: Any isometric immersion $\pmb{f}_1$ of a connected open set $U (\subset P^2(\pmb{Cay}))$ into $\pmb{R}^{26}$ coincides with $\pmb{f}_0$ up to a euclidean transformation of $\pmb{R}^{26}$, i.e., there is a euclidean transformation $a$ of $\pmb{R}^{26}$ satisfying $\pmb{f}_1=a\pmb{f}_0$ on $U$.

MSC(Primary)53C24
MSC(Secondary)53C35, 53B25, 17B20
Uncontrolled Keywordscurvature invariant, isometric immersion, Cayley projective plane, rigidity