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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)

##### MSC:
 53C35 Differential geometry of symmetric spaces
##### Keywords:
rank-one symmetric space; Jacobi operator; eigenvalues
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