Abstract.
For a Riemannian manifold Mn with the curvature tensor R, the Jacobi operator R X is defined by R X Y=R(X,Y)X. The manifold Mn is called pointwise Osserman if, for every p ∈ Mn, the eigenvalues of the Jacobi operator R X do not depend of a unit vector X ∈ T p Mn, and is called globally Osserman if they do not depend of the point p either. R. Osserman conjectured that globally Osserman manifolds are flat or locally rank-one symmetric. This Conjecture is true for manifolds of dimension n≠8,16[14]. Here we prove the Osserman Conjecture and its pointwise version for 8-dimensional manifolds.
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Mathematics Subject Classification (2000): 53B20
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Nikolayevsky, Y. Osserman manifolds of dimension 8. manuscripta math. 115, 31–53 (2004). https://doi.org/10.1007/s00229-004-0480-y
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DOI: https://doi.org/10.1007/s00229-004-0480-y