Hokkaido Mathematical Journal

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

Yoshio AGAOKA and Eiji KANEDA

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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 $U$.

Article information

Source
Hokkaido Math. J., Volume 34, Number 2 (2005), 331-353.

Dates
First available in Project Euclid: 29 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1285766226

Digital Object Identifier
doi:10.14492/hokmj/1285766226

Mathematical Reviews number (MathSciNet)
MR2159001

Zentralblatt MATH identifier
1129.53038

Subjects
Primary: 53C24: Rigidity results
Secondary: 53C35: Symmetric spaces [See also 32M15, 57T15] 53B25: Local submanifolds [See also 53C40] 17B20: Simple, semisimple, reductive (super)algebras

Keywords
curvature invariant isometric immersion Cayley projective plane rigidity

Citation

AGAOKA, Yoshio; KANEDA, Eiji. Rigidity of the canonical isometric imbedding of the Cayley projective plane $P^2(Cay)$. Hokkaido Math. J. 34 (2005), no. 2, 331--353. doi:10.14492/hokmj/1285766226. https://projecteuclid.org/euclid.hokmj/1285766226


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