Laplacians and Riemannian submersions with totally geodesic fibres
Laplacians and Riemannian submersions with totally geodesic fibres
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具有完全测地线纤维的拉普拉斯和黎曼淹没
DOI：
10.1215/ijm/1256046790
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发表时间：
1982
影响因子：
0.6
通讯作者：
J. Bourguignon
中科院分区：
文献类型：
--
作者：
L. Bergery;J. Bourguignon
紧黎曼流形（M,）上的拉普拉斯-贝尔特拉米算子可以看作是普通拉普拉斯算子在r中有界域上的自然推广。特别是，后者的特征值和特征函数的大多数性质，无论是Dirichlet边值还是neumann边值，都适用于前者的谱数据。作为一个例子，著名的Faber-Krahn不等式（见[10]，第188页）从R中的一个区域的面积与它的第一个特征值的乘积的下界找到了J. Hersch定理（见[14]）的对应部分，它给出了2球S2上任何黎曼度规的第一个非零特征值与黎曼面积的乘积的上界。Faber-Krahn不等式推广到r中的域。在b[2]中，M. Berger证明，如果坚持标准度规的上界是尖锐的，则对于n球S不存在这样的推广。这还剩下一些。希望有上界存在。类似地，Courant的节点线定理（见[10]，第452页），根据该定理，第i个特征函数有最多的节点域（即其零集补的连通分量）对一般黎曼流形有效。这样的结果表明，高特征值对应的特征函数应该是复杂的。特别是，它们应该有两个以上的节点域。同样地，我们有理由相信黎曼度规的对称性越多，它的特征值就越多重。特别地，期望球面上的标准度规是其第一个非零特征值的多重性最大的度规。对于（M, g）的适当选择，所有这些猜测都是错误的。正确的选择都属于本文所讨论的流形，即具有完全测地线纤维的黎曼浸没。这些指标是继黎曼积之后的下一个简单指标。这可能表明，在某种意义上，紧流形上的黎曼度量形成了一个比R中的定义域更广的族。
0.0 The Laplace-Beltrami operator on a compact Riemannian manifold (M, ) can be viewed as a natural generalization Ofthe ordinary Laplacian on a bounded domain in R. In particular, most properties ofeigenvalues and eigenfunctions of the latter with either Dirichlet or Neuman boundary values carry out to the spectral data of the former. As an example, the famous Faber-Krahn inequality (see [10], page 188) which bounds from below the product of the area of a domain in R by its first eigenvalue finds its counterpart in J. Hersch’s theorem (cf. [14]) which gives an upper bound of the product of the first nonzero eigenvalue of any Riemannian metric on the 2-sphere S2 by the Riemannian area. The Faber-Krahn inequality generalizes to domains in R. In [2], M. Berger shows that such an extension does not exist for the n-sphere S, if one insists on the upper bound being sharp for the standard metric. This still leaves some.hope for the existence of an upper bound. Analogously, Courant’s nodal line theorem (see [10], page 452) according to which the ith eigenfunction has at most nodal domains (i.e., connected components of the complement of its zero set) is valid for a general Riemannian manifold. Such a result suggests that eigenfunctions corresponding to high eigenvalues should be complicated. In particular they were expected to have more than two nodal domains. In the same vein, it is reasonable to believe that the more symmetries a Riemannian metric has, the more multiple its eigenvalues are. In particular, the standard metric on the sphere was expected to be the metric with the largest multiplicity of its first nonzero eigenvalue. All these guesses turn out to be wrong for appropriate choices of (M, g). The right choices all belong to the class ofmanifolds to which this article is devoted, namely, Riemannian Submersions with totally geodesic fibres. These metrics are the next simple metrics after Riemannian products. This probably indicates that, in a sense, Riemannian metrics on compact manifolds form a wider family than domains in R.