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Osserman manifolds of dimension 8. (English) Zbl 1065.53034

Let \((M,g)\) be a connected Riemannian manifold of dimension \(m\). Let \(J(x)\) be the associated Jacobi operator. One says that \((M,g)\) is pointwise Osserman if for every \(P\in M\), the eigenvalues of \(J(\cdot)\) are constant on the unit sphere \(S(T_PM)\) of the tangent space of \(M\) at \(P\); one says that \((M,g)\) is globally Osserman if \((M,g)\) is pointwise Osserman and if the eigenvalues in question do not vary with \(P\). In the present paper, the author shows
Theorem: A pointwise Osserman manifold of dimension 8 is either flat or is locally a rank \(1\)-symmetric space.
Combined with previous work by Chi and Nikolaveysky, this yields the following
Theorem: If \(m\neq 2,4,16\) and if \((M,g)\) is pointwise Osserman, then \((M,g)\) is either flat or is locally a rank \(1\)-symmetric space. If \(m=2,4\) and if \((M,g)\) is globally Osserman, then \((M,g)\) is either flat or is locally a rank \(1\)-symmetric space.
The case \(m=16\) remains open, although there are some results available. The new results in the present paper deal with the case \(m=16\), the cases \(m\neq8,16\) having been treated by earlier work of Q.-S. Chi [J. Differ. Geom. 28, No. 2, 187–202 (1988; Zbl 0654.53053)] and Y. Nikolayevsky [Differ. Geom. Appl. 18, No. 3, 239–253 (2003; Zbl 1051.53039), Houston J. Math. 29, No. 1, 59–75 (2003; Zbl 1069.53042), (math.DG/0204258)].

MSC:

53C20 Global Riemannian geometry, including pinching
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