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On the holonomy group of the normal connection. (English) Zbl 0799.53059
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 587-595 (1992).
It is well known that for a given Riemannian manifold the representation of the restricted holonomy group on the tangent space is a direct product of a trivial representation and irreducible ones. By the author an analogous result was proved for the holonomy group of the normal connection of a submanifold of a space of constant curvature [see: The normal holonomy group, Proc. Am. Math. Soc. 110, No. 3, 813-818 (1990; Zbl 0708.53023)]. In this case the nontrivial factors are isotropy representations of simple Riemannian symmetric spaces. In the paper under review the following result is presented: Every irreducible isotropy representation of a simple Riemannian symmetric space of rank \(\geq 2\) can arise in this way from an appropriate submanifold of a sphere with the possible exception of 10 representations which are listed. A detailed proof is given by E. Heintze and the author in Indiana Univ. Math. J. 41, No. 3, 869-874 (1992; Zbl 0759.53034).
For the entire collection see [Zbl 0764.00002].
MSC:
53C35 Differential geometry of symmetric spaces
53C40 Global submanifolds
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