An ambient approach to conformal geodesics
An ambient approach to conformal geodesics
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共形测地线的环境方法
DOI：
10.1142/s0219199721500097
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发表时间：
2019
影响因子：
1.6
通讯作者：
Yannick Herfray
中科院分区：
文献类型：
--
作者：
J. Fine;Yannick Herfray
共形测地线是共形流形上的特殊曲线，大致类似于黎曼几何中的测地线。它们的一个定义是作为由共形结构确定的三阶微分方程的解。有一种通过拖拉机演算的替代描述。在这篇文章中，我们使用全息术的思想给出第三种描述。共形[公式：[公式：见正文]-流形[公式：见正文]可以被看作（至少形式上）庞加莱-爱因斯坦[公式：见正文]-流形[公式：见正文]的渐近边界。我们证明[Formula：see text]中的任何曲线[Formula：see text]都有一个唯一确定的延伸到[Formula：see text]中的曲面[Formula：see text]，我们称之为[Formula：see text]的环境曲面。该曲面与边界[公式：见正文]沿着[公式：见正文]成直角相交，并根据其是重整化面积的临界点的要求而被挑选出来。[Formula：see text]的共形几何被编码在[Formula：see text]的黎曼几何中。特别地，[式：当[公式：见正文]是渐近全测地线时，[公式：见正文]是共形测地线，即它的第二基本形式在比预期高一阶时消失。我们还将这种结构与拖拉机和弗曼和格雷厄姆的环境度量结构联系起来。在[Formula：see text]维环境流形中，环境曲面是标度丛上的一个图。于是，沿着沿着这个曲面，牵引器演算就等同于通常的张量演算。这给出了我们共形测地线的全息刻画的另一个紧凑的证明。
Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third-order differential equation determined by the conformal structure. There is an alternative description via the tractor calculus. In this article, we give a third description using ideas from holography. A conformal [Formula: see text]-manifold [Formula: see text] can be seen (formally at least) as the asymptotic boundary of a Poincaré–Einstein [Formula: see text]-manifold [Formula: see text]. We show that any curve [Formula: see text] in [Formula: see text] has a uniquely determined extension to a surface [Formula: see text] in [Formula: see text], which we call the ambient surface of [Formula: see text]. This surface meets the boundary [Formula: see text] in right angles along [Formula: see text] and is singled out by the requirement that it be a critical point of renormalized area. The conformal geometry of [Formula: see text] is encoded in the Riemannian geometry of [Formula: see text]. In particular, [Formula: see text] is a conformal geodesic precisely when [Formula: see text] is asymptotically totally geodesic, i.e. its second fundamental form vanishes to one order higher than expected. We also relate this construction to tractors and the ambient metric construction of Fefferman and Graham. In the [Formula: see text]-dimensional ambient manifold, the ambient surface is a graph over the bundle of scales. The tractor calculus then identifies with the usual tensor calculus along this surface. This gives an alternative compact proof of our holographic characterization of conformal geodesics.
CR 不变量的环境度量构造
DOI：
--
发表时间：
2007
期刊：
IMA in Math.and Appl. 144
影响因子：
--
作者：
M.E.Dokukin;K.Togo;and M.Inoue;武田 三男;K. Hirachi
通讯作者：
K. Hirachi