关于SO(3)上双不变度量与双不变距离的唯一性、关联及相关问题的技术问询

Hey there! Let's work through your questions one by one—this is a super common point of confusion when first diving into Lie groups and Riemannian geometry, so you're not alone here.

问题1：Is any distance on a Lie group possibly induced by some Riemannian metric? #

Short answer: No. Not every topological distance function (what mathematicians call a "metric" in the topological sense) comes from a Riemannian metric on the Lie group.

Riemannian metrics induce geodesic distances, which have strict properties that not all distance functions satisfy:

• They behave like Euclidean distance locally (in small neighborhoods around any point, the distance matches up to scaling with standard Euclidean distance in local coordinates).
• For any two sufficiently close points, there's a path (a geodesic) whose length exactly equals the distance between them.
• They're smooth away from the diagonal (the squared distance function is smooth when points are not identical, in local coordinates).

For example, a discrete distance (where d(p,q)=0 if p=q, 1 otherwise) can't be induced by any Riemannian metric, since it doesn't have the local Euclidean behavior.

问题2：Why are there at least two bi-invariant distances on $SO(3)$? #

This comes down to a critical distinction: the uniqueness statement refers to bi-invariant Riemannian metrics (up to scale), not bi-invariant topological distance functions. Let's break this down with your examples:

• Angular distance $d_1(R_1,R_2) = \arccos (\frac{Tr(R_1^TR_2)-1}{2})$: This is the geodesic distance induced by the unique (up to scale) bi-invariant Riemannian metric on $SO(3)$. The metric comes from the Killing form on the Lie algebra $\mathfrak{so}(3)$ (scaled appropriately), and geodesics on $SO(3)$ are exactly the one-parameter rotation subgroups—so the distance between two rotations is the smallest angle you need to rotate to get from one to the other.
• Chordal distance $d_2(R_1,R_2) = ||R_1 - R_2||_F$: This is the ambient Euclidean distance from the space of all 3x3 matrices (viewed as $\mathbb{R}^9$) restricted to $SO(3)$. While it's bi-invariant (since orthogonal transformations preserve the Frobenius norm), it's not the geodesic distance induced by any Riemannian metric on $SO(3)$. The chordal distance cuts straight through the ambient Euclidean space, not following paths that stay on the $SO(3)$ manifold.

Even though the Frobenius inner product restricted to the tangent spaces of $SO(3)$ is a bi-invariant Riemannian metric (and a scalar multiple of the Killing form metric), the chordal distance isn't the geodesic distance from that metric—it's just the ambient space's distance. That's why you have two distinct bi-invariant distances: one is the manifold's intrinsic geodesic distance, the other is an extrinsic distance from the ambient space.

问题3：How can I tell if a distance is induced by a Riemannian metric? #

Look for these key properties:

• Local Euclidean behavior: In small neighborhoods around any point, the distance should be equivalent to Euclidean distance (when using local coordinates on the manifold). If the distance has "sharp corners" or doesn't match Euclidean scaling locally, it's not Riemannian.
• Geodesic realization: For any two nearby points, there must exist a path lying entirely on the manifold whose length equals the distance between them (this path is a geodesic). Global realization might fail for non-complete manifolds, but local realization is a must.
• Smoothness: The squared distance function should be smooth when evaluated on pairs of distinct points (in local coordinates). This is a consequence of the Riemannian metric being smooth.
• Avoid ambient shortcuts: If the distance is defined as the straight-line distance in an ambient Euclidean space, it's only the Riemannian geodesic distance if the manifold is a linear subspace of that space (which $SO(3)$ is not—it's a curved submanifold).

初学者书籍推荐 #

Lie Groups #

• Lie Groups, Lie Algebras, and Representations by Brian Hall: Hands down the best starting point for beginners. It's intuitive, packed with concrete examples (including $SO(3)$), and avoids overly abstract jargon until you're ready for it.
• Elementary Lie Group Theory by Anthony Knapp: A short, concise text that focuses on the core basics of Lie groups with minimal fluff—great if you want a quick, focused intro.

Riemannian Geometry #

• Riemannian Manifolds: An Introduction to Curvature by John Lee: The standard beginner's textbook for Riemannian geometry. It covers all the basics (geodesics, affine connections, curvature) with clear explanations and practice problems.
• Spacetime and Geometry by Sean Carroll: While focused on general relativity, its first few chapters give an incredibly intuitive introduction to Riemannian geometry that's perfect if you're coming from a physics or applied math background (which is common when working with $SO(3)$).

备注：内容来源于stack exchange，提问作者Risss