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The normal holonomy group. (English) Zbl 0708.53023
It is shown that the restricted normal holonomy group of a submanifold of a space of constant curvature is compact and that the nontrivial part of its representation on the normal space is the isotropy representation of a semisimple Riemannian symmetric space. The proof is based on methods developed in a paper of J. Simons [Ann. Math., II. Ser. 76, 213-234 (1962; Zbl 0106.15201)] and a transformation of the normal curvature tensor to a curvature-like tensor with normal vector fields as entries which uses the shape operators and preserves the geometric information on the normal curvature.
Reviewer: Bernd Wegner

MSC:
53B25 Local submanifolds
53C40 Global submanifolds
53C35 Differential geometry of symmetric spaces
53C29 Issues of holonomy in differential geometry
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References:
[1] Marcel Berger, Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955), 279 – 330 (French). · Zbl 0068.36002
[2] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. · Zbl 0091.34802
[3] James Simons, On the transitivity of holonomy systems, Ann. of Math. (2) 76 (1962), 213 – 234. · Zbl 0106.15201
[4] C. Olmos and C. Sanchez, A geometric characterization of \( R\)-spaces, preprint.
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