The ambient obstruction tensor and the conformal deformation complex

The ambient obstruction tensor and the conformal deformation complex
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环境障碍张量和共形变形复合体

DOI：
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发表时间：
2004
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通讯作者：
Lawrence J. Peterson
Lawrence J. Peterson
中科院分区：
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文献类型：
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作者：
Rod A. Gover;Lawrence J. Peterson

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我们在这里构建一个共形不变的微分算子的代数Weyl张量，使特殊的弯曲类似物的某些运营商有关的变形复杂的，并在应用程序的Weyl曲率，产生（Favorman？环境阻塞张量。这个新的定义的阻塞张量导致简单的直接证明，阻塞张量是发散自由的，消失相同的共形爱因斯坦度量。我们的主要建设的基础上的环境度量的弗罗曼？格雷厄姆及其与保形牵引联络的关系。我们证明了，障碍张量是一个障碍，找到一个环境度量曲率调和一定（环境）形式的拉普拉斯算子。这导致了一个新的环境公式的障碍，在权力的这种形式拉普拉斯作用于环境曲率。这一结果使我们能够构建拉普拉斯型运营商，推广共形拉普拉斯的格雷厄姆？詹尼？梅森Sparling.我们给出了一个算法来计算这些运营商的明确的公式，这是适用于给公式的障碍张量在6和8维。作为这些问题的背景，我们给出了一个明确的结构的变形复杂的尺寸n = 4，构建两个相关的（迂回）复杂的，并建立在这些运营商的基本属性。
We construct here a conformally invariant differential operator on algebraic Weyl tensors that gives special curved analogues of certain operators related to the deformation complex and that, upon application to the Weyl curvature, yields the (Fefferman?Graham) ambient obstruction tensor. This new definition of the obstruction tensor leads to simple direct proofs that the obstruction tensor is divergence-free and vanishes identically for conformally Einstein metrics. Our main constructions are based on the ambient metric of Fefferman?Graham and its relation to the conformal tractor connection. We prove that the obstruction tensor is an obstruction to finding an ambient metric with curvature harmonic for a certain (ambient) form Laplacian. This leads to a new ambient formula for the obstruction in terms of a power of this form Laplacian acting on the ambient curvature. This result leads us to construct Laplacian-type operators that generalise the conformal Laplacians of Graham?Jenne?Mason?Sparling. We give an algorithm for calculating explicit formulae for these operators, and this is applied to give formulae for the obstruction tensor in dimensions 6 and 8. As background to these issues, we give an explicit construction of the deformation complex in dimensions n = 4, construct two related (detour) complexes, and establish essential properties of the operators in these.