Curvature characterization and classification of rank-one symmetric spaces. (English) Zbl 0742.53018

For a compact rank-one symmetric space (not of constant curvature) the Jacobi operator \(K_ v=R(\cdot,v)v\) (\(R\) is the curvature tensor, \(v\) a unit vector) has the following two properties: (1) \(K_ v\) has two distinct constant eigenvalues 1 and \(1/4\); (2) for \(E_ 1(v)\), spanned by \(v\) and the 1-eigenspace of \(K_ v\), we have: \(E_ 1(w)=E_ 1(v)\) whenever \(w\) is in \(E_ 1(v)\). The main theorem states that a locally rank-one symmetric space not of constant curvature is characterized by the two properties (1) and (2). Based on these interesting geometric considerations, the author gives a Lie algebra classification of the rank-one compact symmetric spaces.
Reviewer: V.Boju (Craiova)


53C35 Differential geometry of symmetric spaces
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