Cones over pseudo-Riemannian manifolds and their holonomy
Cones over pseudo-Riemannian manifolds and their holonomy
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伪黎曼流形上的锥及其完整性
DOI：
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发表时间：
2007
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通讯作者：
T. Leistner
中科院分区：
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作者：
D. Alekseevsky;V. Cortés;A. Galaev;T. Leistner
摘要 根据 Gallot 的经典定理 (Ann. Sci. Éc. Norm. Sup. 12: 235–267, 1979)，完全黎曼流形上的黎曼锥要么是平坦的，要么具有不可约完整性。我们考虑伪黎曼流形上具有可约完整的度量锥体。首先我们描述当圆锥完整可分解时圆锥底部的局部结构。例如，我们发现基的完整代数始终是完全伪正交李代数。全局结果之一是紧致完备伪黎曼流形上的圆锥要么是平坦的，要么具有不可分解的完整性。然后我们分析锥体具有不可分解但可约的完整性的情况，这意味着它承认平行各向同性分布。首先在锥体允许两个互补分布的情况下进行该分析，其次针对洛伦兹锥体进行分析。我们证明，当基流形的局部几何形状是对 Sasakian 且圆锥体的局部几何形状是对 Kahlerian 时，第一种情况恰好发生。对于洛伦兹锥，我们根据基流形的度量得到了可能的（局部）完整代数的完整描述。
Abstract By a classical theorem of Gallot (Ann. Sci. Éc. Norm. Sup. 12: 235–267, 1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is decomposable. For instance, we find that the holonomy algebra of the base is always the full pseudo-orthogonal Lie algebra. One of the global results is that a cone over a compact and complete pseudo-Riemannian manifold is either flat or has indecomposable holonomy. Then we analyse the case when the cone has indecomposable but reducible holonomy, which means that it admits a parallel isotropic distribution. This analysis is carried out, first in the case where the cone admits two complementary distributions and, second for Lorentzian cones. We show that the first case occurs precisely when the local geometry of the base manifold is para-Sasakian and that of the cone is para-Kählerian. For Lorentzian cones we get a complete description of the possible (local) holonomy algebras in terms of the metric of the base manifold.