ON OSSERMAN MANIFOLDS OF DIMENSION 16 ∗
ON OSSERMAN MANIFOLDS OF DIMENSION 16 ∗
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尺寸 16 的 OSSERMAN 歧管 *
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
Y. Nikolayevsky
中科院分区：
文献类型：
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作者：
Y. Nikolayevsky
对于曲率张量为R的黎曼流形Mn，Jacobi算子RX定义为RXY = R（X，Y）X。流形Mn称为点态Osserman，如果对于每个p ∈ Mn，Jacobi算子RX的特征值不依赖于单位向量X ∈ TpM的选择，并且如果它们也不依赖于点p，则称为全局Osserman。R. Osserman证明了全局Osserman流形是平坦的或秩一对称的。除了n = 16维流形的Jacobi算子有重数m ∈ {7，8，9}的特征值外，这个猜想在所有情况下都得到了证明。这里我们给出m = 9时的证明。
For a Riemannian manifold Mn with the curvature tensor R, the Jacobi operator RX is defined by RXY = R(X, Y )X. The manifold M n is called pointwise Osserman if, for every p ∈ Mn, the eigenvalues of the Jacobi operator RX do not depend on the choice of a unit vector X ∈ TpM, 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 rank-one symmetric. This Conjecture was proved in all the cases, except for manifolds of dimension n = 16 whose Jacobi operator has an eigenvalue of multiplicity m ∈ {7, 8, 9}. Here we give the proof in the case m = 9.