## Rigidity of the canonical isometric imbedding of the quaternion projective plane $$P^2(\mathbb H)$$.(English)Zbl 1102.53042

Summary: We investigate isometric immersions of $$P^2(\mathbb H)$$ into $$\mathbb R^{14}$$ and prove that the canonical isometric imbedding $${\mathbf f}_0$$ of $$P^2(\mathbb H)$$ into $$\mathbb R^{14}$$, which is defined by S. Kobayashi [Tohoku Math. J., II. Ser. 20, 21–25 (1968; Zbl 0175.48301)], is rigid in the following strongest sense: Any isometric immersion $${\mathbf f}_1$$ of a connected open set $$U$$ $$(\subset P^2(\mathbb H))$$ into $$\mathbb R^{14}$$ coincides with $${\mathbf f}_0$$ up to an Euclidean transformation of $$\mathbb R^{14}$$, i.e., there is an Euclidean transformation $$a$$ of $$\mathbb R^{14}$$ satisfying $${\mathbf f}_1= a{\mathbf f}_0$$ on $$U$$.

### MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C24 Rigidity results 17B20 Simple, semisimple, reductive (super)algebras

Zbl 0175.48301
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