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].

For the entire collection see [Zbl 0764.00002].

Reviewer: H.Reckziegel (Köln)

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\textit{C. Olmos}, in: 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; Budapest: János Bolyai Mathematical Society. 587--595 (1992; Zbl 0799.53059)