奇Pfidian形式
Odd Pfaffian forms
摘要：
在任何奇维定向黎曼流形上，我们通过一个在曲率张量中具有整系数的不变多项式定义了一个体积形式，我们称之为奇黎曼。我们证明了一个内在的Chern-Gauss-Bonnet公式的不完整的边缘奇异性的奇Pfronan的纤维的边界纤维。该公式适用于产品型模型的边缘度量的退化是圆锥型在每个纤维，但也为一般类的扰动的模型度量。同样的方法产生了一个陈-高斯-博内公式的完整，非紧流形的纤维边界的意义上的Mazzeo-Melrose和扰动，涉及奇Pfronan的基础纤维化。我们推导出的合理性，通常Pfavian形式的黎曼orbifolds，并表现出障碍的某些度量纤维化实现的模型在无穷远的平面度量与圆锥形，边缘或纤维边界奇点。
英文摘要：
On any odd-dimensional oriented Riemannian manifold we define a volume form, which we call the odd Pfaffian, through a certain invariant polynomial with integral coefficients in the curvature tensor. We prove an intrinsic Chern-Gauss-Bonnet formula for incomplete edge singularities in terms of the odd Pfaffian on the fibers of the boundary fibration. The formula holds for product-type model edge metrics where the degeneration is of conical type in each fiber, but also for general classes of perturbations of the model metrics. The same method produces a Chern- Gauss-Bonnet formula for complete, non-compact manifolds with fibered boundaries in the sense of Mazzeo-Melrose and perturbations thereof, involving the odd Pfaffian of the base of the fibration. We deduce the rationality of the usual Pfaffian form on Riemannian orbifolds, and exhibit obstructions for certain metrics on a fibration to be realized as the model at infinity of a flat metric with conical, edge or fibered boundary singularities.
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