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.

翻译：黎曼度量的常数重缩放出现在许多计算场景中，通常通过显式或隐式引入的全局尺度参数实现。尽管这一操作是基本的，但其实际影响在实践中并不总是明确的，且可能与曲率变化、流形结构或坐标表示的变化相混淆。本文简要、独立地阐述了任意黎曼流形上的常数度量缩放。我们区分了在此类缩放下发生变化的量——包括范数、距离、体积元和梯度幅度——以及保持不变的几何对象，例如列维-奇维塔联络、测地线、指数映射与对数映射以及平行移动。我们还讨论了其对黎曼优化的影响，其中常数度量缩放通常可解释为步长的全局重缩放，而非对基础几何的修改。本文的目标纯粹是阐述性的，旨在阐明如何在黎曼计算中引入全局度量尺度参数，而不改变这些方法所依赖的几何结构。