Rank rigidity and symmetry. (Q2760472)

The paper contains a broad study of Riemannian symmetric spaces of higher rank. The author considers a class of theorems that characterize several different aspects of these spaces such as: rank rigidity, Desarguesian property, Riemannian manifolds of nonpositive curvature, spherical Tits buildings, infinitesimal rank rigidity, isoparametric submanifolds. Tits generalized the Desarguesian property by constructing the analog of the transformations [see \textit{J. Tits}, Buildings of spherical type and finite BN-pairs, Springer Lect. Notes Math. 386 (1974; Zbl 0295.20047)]. He proved that if a locally connected irreducible compact spherical building has rank \(r\geq3\), then it is associated to a symmetric space of rank \(r\). J.-H. Eschenburg gives a new prove of Thorbergsson's result that a closed, irreducible, substantial isoparametric submanifold of codimension \(k\geq3\) in Euclidean \(n\)-space is a principal orbit of the isotropy representation of a symmetric space of rank \(k\) [see \textit{G. Thorbergsson}, Ann. Math. (2) 133, No. 2, 429--446 (1991; Zbl 0727.57028)]. Eschenburg does not use the classification of Dadok used previously by Thorbergsson, instead he uses submanifold theory and some ideas of Olmos [see \textit{C. Olmos}, J. Differ. Geom. 38, 225--234 (1993; Zbl 0791.53051)].