HARMONIC MORPHISMS WITH ONE-DIMENSIONAL FIBRES
HARMONIC MORPHISMS WITH ONE-DIMENSIONAL FIBRES
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DOI：
10.1142/s0129167x99000197
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发表时间：
1999-06
影响因子：
0.6
通讯作者：
R. Pantilie
中科院分区：
文献类型：
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作者：
R. Pantilie
我们通过将调和态射置于共角叶状结构的背景下来研究它们。我们获得的大多数结果都适用于一维纤维和不等于二维的共域。我们考虑在紧致和非紧致黎曼流形上产生调和态射的叶状结构。通过使用积分公式，我们证明了对一维叶状结构的扩展，它产生了 S. Bochner 关于具有非正 Ricci 曲率的紧黎曼流形上的 Killing 场的著名结果的调和态射。从非紧的情况，我们改进了 R. L. Bryant[9] 关于调和态射的结果，该一维纤维定义在至少四维黎曼流形上，具有恒定的截面曲率。我们的方法为科比的结果提供了全新的几何证明。我们引入的同位叶化（或者更一般地说，同位分布）的概念在证明和提供具有任何维度的纤维的调和态射的新示例方面似乎都是一个有用的工具。
We study harmonic morphisms by placing them into the context of conformal foliations. Most of the results we obtain hold for fibres of dimension one and codomains of dimension not equal to two. We consider foliations which produce harmonic morphisms on both compact and noncompact Riemannian manifolds. By using integral formulae, we prove an extension to one-dimensional foliations which produce harmonic morphisms of the well-known result of S. Bochner concerning Killing fields on compact Riemannian manifolds with nonpositive Ricci curvature. From the noncompact case, we improve a result of R. L. Bryant[9] regarding harmonic morphisms with one-dimensional fibres defined on Riemannian manifolds of dimension at least four with constant sectional curvature. Our method gives an entirely new and geometrical proof of Bryant's result. The concept of homothetic foliation (or, more generally, homothetic distribution) which we introduce, appears as a useful tool both in proofs and in providing new examples of harmonic morphisms, with fibres of any dimension.