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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

Citations:

Zbl 0175.48301
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