zbMATH — the first resource for mathematics

A geometric classification of positively curved symmetric spaces and the isoparametric construction of the Cayley plane. (English) Zbl 0669.53038
On the geometry of differentiable manifolds, Workshop, Rome/Italy 1986, Astérisque 163/164, 111-135 (1988).
[For the entire collection see Zbl 0666.00013.]
The paper contains a very beautiful geometric classification of compact rank one symmetric spaces. Jacobi field techniques are used to show that they behave like projective planes, and that the division algebra can be read directly from the geometry. A proof of the classification of norm- preserving real division algebras is included. Then the Cayley plane is constructed as the focal manifold of an isoparametric family on \(S^{25}\). Finally its curvature is computed.
Reviewer: D.Ferus

53C35 Differential geometry of symmetric spaces
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)