Harmonic analysis in vector bundles associated to the rotation and spin groups
Harmonic analysis in vector bundles associated to the rotation and spin groups
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与旋转和自旋群相关的向量束的调和分析
DOI：
10.1016/0022-1236(92)90050-s
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发表时间：
1992
影响因子：
1.7
通讯作者：
T. Branson
中科院分区：
文献类型：
--
作者：
T. Branson
令 V 为球体 Sn 上的任何向量丛，它与定向正交框架的主丛或自旋框架的主丛相关联。我们给出了具有多重性的 Bochner Laplacian ▽*▽ onV 谱的显式公式，其中 ▽ 是黎曼或黎曼自旋连接。对于张量丛，我们还给出了 Lichnerowicz Laplacian 的谱。研究了自然一阶微分算子对相应特征空间的影响，并通过对微分形式丛、旋量、无迹对称张量和代数 Weyl 张量的详细处理来说明所有结果。
LetVbe any vector bundle over the sphereSnwhich is associated to the principal bundle of oriented orthonormal frames, or to that of spin frames. We give an explicit formula for the spectrum, with multiplicity, of the Bochner Laplacian ▽∗▽ onV, where ▽ is the Riemannian or Riemannian spin connection. In the case of tensor bundles, we also give the spectrum of the Lichnerowicz Laplacian. The effect of natural first-order differential operators on the corresponding eigenspaces is investigated, and all results are illustrated in a detailed treatment of some examples: the bundles of differential forms, spinors, trace-free symmetric tensors, and algebraic Weyl tensors.