Abstract DeTurck and Yang have shown that in the neighborhood of every point of a 3-dimensional Riemannian manifold, there exists a system of orthogonal coordinates (that is, with respect to which the metric has diagonal form). We show that this property does not generalize to higher dimensions. In particular, the complex projective spaces CP m and the quaternionic projective spaces HP q , endowed with their canonical metrics, do not have local systems of orthogonal coordinates for m , q ≥ 2 .

中文翻译：

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复数和四元数射影空间上不存在正交坐标

摘要 DeTurck 和Yang 已经证明，在3 维黎曼流形的每个点的邻域中，都存在一个正交坐标系（即，度量具有对角线形式）。我们表明这个属性不能推广到更高的维度。特别是，复射影空间 CP m 和四元数射影空间 HP q 赋予了它们的规范度量，没有 m 的局部正交坐标系，q ≥ 2。