三维可解李群上的适配复极化
Adapted complex polarizations on solvable Lie groups of dimension 3
- ELTE - Eötvös L. Univ.（罗兰大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究实李群上左不变Koszul联络相关的适配复极化，利用联络测地流的$i\boldsymbol R$-完备性判据，给出李群切丛上左不变ac-极化实例，证明了若干非局部对称但$\boldsymbol P$-完备的联络。
AI中文摘要：
本文研究与实李群上左不变Koszul联络相关的适配复极化，适配复极化是流形上复结构的自然推广。我们研究特定李群的切丛上这类极化的存在性，该李群带有左不变联络。我们依据联络测地流的$i\boldsymbol R$-完备性，给出全局适配复极化存在的实用判据。在李群情形中，Euler-Arnold向量场决定测地流的行为。结合特定李群的Euler-Arnold向量场相关知识与$i\boldsymbol R$-完备性判据，我们给出若干定义在李群切丛上的左不变ac-极化实例。除伪黎曼度量联络外，我们还在若干情形中讨论双不变联络。我们的研究聚焦于可解李群，因为非可解李群已被更充分地理解。本文主要结果是，我们证明了若干非局部对称但$\boldsymbol P$-完备的联络。
英文摘要：
In this paper we investigate adapted complex polarizations associated with left-invariant Koszul connections on real Lie groups. Adapted complex polarizations arise as a natural generalization of a complex structure on a manifold. We investigate the existence of such polarizations on the tangent bundle of certain Lie groups with left-invariant connections. We use a practical criterion for the existence of global adapted complex polarizations in terms of the $i\mathbb R$-completeness of the geodesic flow of the connection. In the Lie case, the Euler-Arnold vector field determines the behaviour of the geodesic flow. By combining what we know about the Euler-Arnold vector field of certain Lie groups and the $i\mathbb R$-completeness criterion, we provide several examples of left-invariant ac-polarizations defined on the tangent bundle of Lie groups. Along with pseudo-Riemannian metric connections, we discuss biinvariant connections in several cases. Our focus is on solvable Lie groups, because non-solvable Lie groups are better understood. The main result of this paper is that we demonstrate several not locally symmetric but $\mathcal P$-complete connections.
