frame bundle of an n-sphere is $O(n+1)$?

Question

I'm trying to understand frame bundles (mostly from a physics point of view). It is standard for some n-dimensional Riemannian manifold that the frame bundle has an $O(n)$ structure. However; It seems there is some exception for an n-sphere where I come across references to it's oriented-orthonormal frame bundle being diffeomorphic to $SO(n+1)$. In such cases they say that the matrix with columns $\left[\phi,\hat{e}_{i}...\hat{e}_{n}\right]\in SO(n+1)$. Where $\phi$ are the coordinates of the n-sphere as embedded within $\mathbb{R}^{n+1}$ and the hat's are the unit frame vectors.

Can someone please elaborate upon this for me, I don't quite get it, does this only apply to a sphere of constant curvature? I was trying to construct an explicit example for the 3-sphere but couldn't quite get it. Does this apply to the general topological n-sphere or just a constant curvature one?

Answer 1

There are two issues here:

1. The orthonormal frame bundle of a Riemannian $n$-manifold has structure group $O(n)$.
2. The total space of the oriented orthonormal frame bundle of a round $n$-sphere may be identified with $SO(n+1)$: As you say, an element of $SO(n+1)$ with first column $\phi$ is naturally identified with an orthonormal frame of $T_{\phi} S^{n}$, the tangent space at $\phi$ to the round unit sphere in Euclidean $(n+1)$-space.

For a non-round Riemannian manifold diffeomorphic to a sphere, the orthonormal frame bundle is diffeomorphic to $SO(n+1)$, but not identified with $SO(n+1)$ in the same literal way.

(Separately, take care when speaking of "topological" spheres. In dimensions starting with $7$, there exist exotic spheres, smooth manifolds homeomorphic to the sphere but not diffeomorphic to the sphere.)