Isometric actions and harmonic morphisms.
Isometric actions and harmonic morphisms.
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DOI：
10.1307/mmj/1030132589
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发表时间：
2000
影响因子：
0.9
通讯作者：
R. Pantilie
中科院分区：
文献类型：
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作者：
R. Pantilie
众所周知，具有最小叶的黎曼叶理具有产生调和态射的性质，即它的叶是淹没调和态射的局部纤维。这是具有最小纤维的黎曼淹没是谐波态射这一事实的直接结果。更一般地说，黎曼叶理（余维不等于2）产生调和态射当且仅当由叶的平均曲率决定的向量场局部是一个梯度向量场。这是Baird和Eells的基本方程的结果（见下一节的命题1.2）。虽然这个条件很简单，但已知的黎曼叶理例子很少；我们的工作将提供许多新的。对于一维黎曼叶理，上述条件等价于叶理是局部由杀戮场产生的事实（由于Bryant[6]的结果），但对于大于1维的叶理就不成立了。本文证明了对于由杀戮场局部生成的叶理，其条件仅取决于水平分布的可积张量和诱导的局部作用。由此得到了由杀伤场局部生成的叶理产生调和态射的一个有用判据。这在第1节（定理1.13）中完成。在第2节中，我们推导了一些结果，从而得到了产生调和态射的以下类黎曼叶（余维数为6= 2）：
It is well known that a Riemannian foliation with minimal leaves has the property that it produces harmonic morphisms, that is, it leaves are locally fibers of submersive harmonic morphisms. This is an immediate consequence of the fact that Riemannian submersions with minimal fibers are harmonic morphisms. More generally, a Riemannian foliation (of codimension not equal to 2) produces harmonic morphisms if and only if the vector field determined by the mean curvatures of the leaves is locally a gradient vector field. This is a consequence of the fundamental equation of Baird and Eells [1] (see Proposition 1.2 in the next section). Although this condition is quite simple, few examples of such Riemannian foliations were known; our work will provide many new ones. For a 1-dimensional Riemannian foliation, the condition just stated is equivalent to the fact that the foliation is locally generated by Killing fields (a result due to Bryant [6]), but this is not true for foliations of dimension greater than 1. In this paper we show that, for a foliation locally generated by Killing fields, the condition depends only on the integrability tensor of the horizontal distribution and the induced local action. Thus we obtain a useful criterion for a foliation locally generated by Killing fields to produce harmonic morphisms. This is done in Section 1 (Theorem1.13). In Section 2 we derive a few consequences, thus obtaining the following classes of Riemannian foliations (of codimension 6= 2) that produce harmonic morphisms: