论文标题
当地共同产品结构
Locally conformally product structures
论文作者
论文摘要
紧凑型歧管$ M $上的本地保质产品（LCP）结构是一种保形结构$ c $，并带有封闭的，非脱位和非载液Weyl Connection $ d $，并具有可还原的自由度。同等地，$ m $上的LCP结构是由可简化的，非牌，不完整的Riemannian Metric $ H_D $ h_d $ the Universal Cover $ \ tilde m $ $ m $ $ $ $ $ $的，与基本组$π_1（m）$相似。 Kourganoff最近证明，在这种情况下，$（\ tilde m，h_d）$是对平面空间$ \ mathbb {r}^q $的Riemannian产品的等值线，并且不完整的不完全不可记录的Riemannian歧管歧管$（N，G_N）$。 In this paper we show that for every LCP manifold $(M,c,D)$, there exists a metric $g\in c$ such that the Lee form of $D$ with respect to $g$ vanishes on vectors tangent to the distribution on $M$ defined by the flat factor $\mathbb{R}^q$, and use this fact in order to construct new LCP structures from a given one by taking products.我们还建立了LCP歧管与数字理论之间的联系，并使用它们来构建大量示例，其中包含由Matveev-Nikolayevsky，Kourganoff和Oeljeklaus-Toma构建的所有先前已知的LCP歧管示例。
A locally conformally product (LCP) structure on compact manifold $M$ is a conformal structure $c$ together with a closed, non-exact and non-flat Weyl connection $D$ with reducible holonomy. Equivalently, an LCP structure on $M$ is defined by a reducible, non-flat, incomplete Riemannian metric $h_D$ on the universal cover $\tilde M$ of $M$, with respect to which the fundamental group $π_1(M)$ acts by similarities. It was recently proved by Kourganoff that in this case $(\tilde M, h_D)$ is isometric to the Riemannian product of the flat space $\mathbb{R}^q$ and an incomplete irreducible Riemannian manifold $(N,g_N)$. In this paper we show that for every LCP manifold $(M,c,D)$, there exists a metric $g\in c$ such that the Lee form of $D$ with respect to $g$ vanishes on vectors tangent to the distribution on $M$ defined by the flat factor $\mathbb{R}^q$, and use this fact in order to construct new LCP structures from a given one by taking products. We also establish links between LCP manifolds and number field theory, and use them in order to construct large classes of examples, containing all previously known examples of LCP manifolds constructed by Matveev-Nikolayevsky, Kourganoff and Oeljeklaus-Toma.
