Conformally convariant equations on differential forms
Conformally convariant equations on differential forms
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DOI：
10.1080/03605308208820228
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发表时间：
1982
影响因子：
1.9
通讯作者：
T. Branson
中科院分区：
文献类型：
--
作者：
T. Branson
设 M 为维度 n≥3 的伪黎曼流形。二阶线性微分算子 是 k 形式上的 Laplace-Beltrami 算子的变体（通过对 d*d 和 dd* 项进行不同的加权而获得）与取决于 M 的 Ricci 张量的零阶算子之和，具有显着的共形拟不变性。具体来说，将 M 的共形群的两个多重表示交织在一起。 ] 是 Laplace-Beltrami 算子在函数上通过 M 的标量曲率倍数进行拟不变平移的自然推广，以及 Maxwell 算子在“向量势”形式上的拟不变性的自然推广。Orsted 在 [9]-[11] 中处理了函数上算子的共形拟不变性，并且间接地由微分几何学家研究在紧流形上规定标量曲率的“Yamabe 问题”（参见参考文献 [7]）。这似乎也为物理学家所熟知。
Let M be a pseudo-Riemannian manifold o f dimension n≧3. A second-order linear differential operator , which is the sum of a variant of the Laplace-Beltrami operator on k-forms (obtained by weighting the d∗d and dd∗ terms differently) and a zeroth order operator depending on t he Ricci tensor of M, has remarkable conformal quasi-invariance properties. Specifically, intertwines two mu1tip1ier representations o f the conformal group of M. The ] are natura1 generalizations of the quasi -invariant shift of the Laplace-Beltrami operator on functions by a multiple of the scalar curvature of M, and the Maxwell operator on “vector potentials “ -forms).. The conformal quasi-invariance of the operator on functions was treated by Orsted in [9]-[11], and in-directly by differential geometers studying the “Yamabe problem” of prescribing the scalar curvature on a compact manifold (see [7] for a bibliography) . It also seems to be known to physicists.