统一解释黎曼流形中度量缩放的数学影响，避免误解为几何改变。
Constant Metric Scaling in Riemannian Computation
- 区分缩放前后哪些量变、哪些不变，如范数变而测地线不变
- 发现全局步长缩放等价于度量缩放，不改变几何结构
- 适合做黎曼优化或几何计算的初学者理解基础概念
在多种计算场景中，常出现对黎曼度量的恒定缩放，通常通过一个全局尺度参数实现，无论是显式还是隐式引入。尽管这一操作看似简单，其实际影响常被忽视或与曲率、流形结构或坐标表示的变化混淆。本文对任意黎曼流形上的恒定度量缩放提供简明自洽的阐述：明确区分缩放后变化的量（如范数、距离、体积元、梯度模长）与保持不变的几何对象（如Levi-Civita联络、测地线、指数映射、对数映射和平行移动）。还讨论了其在黎曼优化中的含义——常可解释为全局步长缩放而非几何修改。本文旨在澄清：引入全局度量缩放参数不会破坏依赖的几何结构。
原文摘要 · Abstract (English)
Constant rescaling of a Riemannian metric appears in many computational settings, often through a global scale parameter that is introduced either explicitly or implicitly. Although this operation is elementary, its consequences are not always made clear in practice and may be confused with changes in curvature, manifold structure, or coordinate representation. In this note we provide a short, self-contained account of constant metric scaling on arbitrary Riemannian manifolds. We distinguish between quantities that change under such a scaling, including norms, distances, volume elements, and gradient magnitudes, and geometric objects that remain invariant, such as the Levi--Civita connection, geodesics, exponential and logarithmic maps, and parallel transport. We also discuss implications for Riemannian optimization, where constant metric scaling can often be interpreted as a global rescaling of step sizes rather than a modification of the underlying geometry. The goal of this note is purely expository and is intended to clarify how a global metric scale parameter can be introduced in Riemannian computation without altering the geometric structures on which these methods rely.
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