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A geometric proof of the Berger holonomy theorem. (English) Zbl 1082.53048
As the author claims, the paper is devoted to a geometric proof of the following holonomy theorem proved by M. Berger in 1955: If the holonomy group of an irreducible Riemannian manifold is not transitive on the sphere, then \(M\) is locally symmetric. The proof (well outlined in the introduction) uses the normal space to an orbit of the holonomy group at a fixed tangent space \(T_pM\) of \(M\). It defines both the totally geodesic normal ball and normal holonomy group, acting as isometries on it. From here it follows that this ball is locally symmetric and is actually part of a family of such balls whose tangent spaces generate the whole \(T_pM\). Then the Jacobi operator is diagonalizable with constant coefficients in a parallel frame, hence \(M\) is locally symmetric. The methods of normal holonomy used in the paper are developed recently by the author and are expected to provide new results in the theory of Riemannian submanifolds of Euclidean spaces.

53C29 Issues of holonomy in differential geometry
53C40 Global submanifolds
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