Chern-Simons不变量与Nil、Sol和Euclidean几何中表示的体积
Chern-Simons invariants and volumes of representations in Nil, Sol, and Euclidean geometries
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中文总结 AI 辅助
本文在Nil、Sol和Euclidean几何中，通过构造辅助联络，将表示的体积实现为Chern-Simons不变量，并恢复黎曼体积。
中文摘要 AI 辅助
本文中，我们将表示的体积实现为Nil、Sol和Euclidean几何中的实值Chern-Simons不变量。为此，我们在一个不必平凡的主丛上，对一对联络构造了Chern-Simons不变量。对于连通的闭定向3流形$M$以及表示$\rho\colon\pi_1(M)\to G$，其中$G$是这些几何之一的等距群的恒等分量，我们在相应的平坦$G$-丛上构造了一个辅助联络。我们证明，对于适当归一化的不变多项式，辅助联络与平坦联络的Chern-Simons不变量等于表示的体积。对于几何结构的和乐表示，该不变量恢复了黎曼体积。我们还计算了每种几何中代表性闭流形的Levi-Civita联络的Chern-Simons不变量。
英文摘要
In this paper, we realize volumes of representations as real-valued Chern-Simons invariants in Nil, Sol, and Euclidean geometries. To this end, we formulate a Chern-Simons invariant of a pair of connections on a principal bundle that need not be trivial. For a connected closed oriented 3-manifold $M$ and a representation $ρ\colonπ_1(M)\to G$ into the identity component $G$ of the isometry group of one of these geometries, we construct an auxiliary connection on the associated flat $G$-bundle. We show that, for a suitably normalized invariant polynomial, the Chern-Simons invariant of the auxiliary and flat connections equals the volume of the representation. For the holonomy representation of a geometric structure, this invariant recovers the Riemannian volume. We also compute the Chern-Simons invariant of the Levi-Civita connection for representative closed manifolds in each of these geometries.
发表机构
- Institute of Science Tokyo（东京科学大学）
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