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 n-manifold X can be seen (formally at least) as the asymptotic boundary of a Poincaré–Einstein (n + 1)-manifold Y. We show that any curve γ in X has a uniquely determined extension to a surface Σγ in Y, which we call the ambient surface ofγ. This surface meets the boundary X in right angles along γ and is singled out by the requirement that it be a critical point of renormalized area. The conformal geometry of γ is encoded in the Riemannian geometry of Σγ. In particular, γ is a conformal geodesic precisely when Σγ 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 (n + 2)-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.

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保形测地线的环境方法

保形测地线是保形流形上的不同曲线，大致类似于黎曼几何的测地线。它们的一个定义是由保形结构确定的三阶微分方程的解。拖拉机微积分有另一种描述。在本文中，我们使用全息术的想法给出第三个描述。保形n-歧管X可以（至少在形式上）被视为 Poincaré-Einstein 的渐近边界(n + 1)-歧管是. 我们证明任何曲线γ在X具有唯一确定的表面延伸Σγ在是，我们称之为环境表面γ. 这个表面符合边界X沿直角γ并以它是重整化区域的临界点的要求而被挑选出来。的保形几何γ用黎曼几何编码Σγ. 尤其，γ是一个保形测地线Σγ是渐近完全测地线的，即它的第二个基本形式消失到比预期高一个数量级。我们还将这种结构与拖拉机和 Fefferman 和 Graham 的环境度量结构联系起来。在里面(n + 2)维环境流形，环境表面是尺度束上的图形。然后，拖拉机演算与沿该表面的通常张量演算一致。这为我们对保形测地线的全息表征提供了另一种紧凑的证明。