Levi–Civita connection as a principal O(n)-connection

Let $(M,g)$ be a Riemannian smooth manifold of dimension $n$ . Denote by $O(M)\to M$ the orthonormal frame bundle, a principal $O(n)$ -bundle.

Theorem (Levi–Civita connection; principal-bundle formulation)

There exists a unique covariant derivative

\nabla\colon \Gamma(TM)\times \Gamma(TM)\to \Gamma(TM)

on the tangent bundle satisfying:

(Metric compatibility) $X\langle Y,Z\rangle = \langle \nabla_XY, Z\rangle + \langle Y,\nabla_XZ\rangle$ for all vector fields $X,Y,Z$ .
(Torsion-free) $\nabla_XY-\nabla_YX = [X,Y]$ , where $[\cdot,\cdot]$ is the Lie bracket of vector fields.

This unique connection is the Levi–Civita connection of $g$ .

Equivalently, there exists a unique principal connection on the principal bundle $O(M)\to M$ whose associated connection on $TM$ (via the defining representation of $O(n)$ ) is $\nabla$ , and whose torsion 2-form (defined using the solder form) vanishes; this torsion is precisely the object defined in torsion 2-form.

Examples

Euclidean space. On $(\mathbb{R}^n,\text{standard }g)$ , $\nabla$ is ordinary differentiation in standard coordinates; the pullback of the connection 1-form along the standard global orthonormal frame is zero.
Round sphere. On $S^n$ with the round metric, $\nabla$ is the tangential component of the ambient derivative in $\mathbb{R}^{n+1}$ ; its holonomy is typically all of $SO(n)$ for $n\ge 2$ .
Bi-invariant metric on a Lie group. If a Lie group carries a bi-invariant metric, then for left-invariant vector fields $X,Y$ one has $\nabla_XY=\tfrac12[X,Y]$ .