Orthonormal frame bundle - Imported Knowlpedia Content

Let $(M,g)$ be an $n$ -dimensional Riemannian manifold. The Riemannian metric gives an inner product $g_x$ on each tangent space $T_xM$ .

The orthonormal frame bundle $O(TM)\to M$ is the subset of the frame bundle consisting of frames that are orthonormal with respect to $g$ :

O(TM)=\{(x,(e_1,\dots,e_n))\in \mathrm{Fr}(TM): g_x(e_i,e_j)=\delta_{ij}\}.

The structure group is the orthogonal group $\mathrm{O}(n)$ (a Lie group), acting on the right by change of orthonormal basis.

With this action, $O(TM)\to M$ is a principal $\mathrm{O}(n)$ -bundle and is a reduction of $\mathrm{Fr}(TM)\to M$ from $\mathrm{GL}(n)$ to $\mathrm{O}(n)$ ; this reduction is the basic example in reducing structure group using a bundle metric.

Examples

Euclidean space. On $M=\mathbb R^n$ with the standard metric, $O(T\mathbb R^n)\cong \mathbb R^n\times \mathrm{O}(n)$ via the constant orthonormal frame.

Oriented Riemannian manifolds. If $M$ is oriented, the orthonormal frame bundle has a natural reduction to the connected subgroup $\mathrm{SO}(n)$ by restricting to oriented orthonormal frames.

Round 2-sphere. For $S^2$ with the round metric, the oriented orthonormal frame bundle can be identified with $\mathrm{SO}(3)$ : an oriented orthonormal tangent frame at $x\in S^2\subset\mathbb R^3$ uniquely extends to an oriented orthonormal frame of $\mathbb R^3$ by adding the outward unit normal.