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

on the tangent bundle satisfying:

1. (Metric compatibility) $X\langle Y,Z\rangle = \langle \nabla_XY, Z\rangle + \langle Y,\nabla_XZ\rangle$ for all vector fields $X,Y,Z$.
2. (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

1. Euclidean space. On $(\mathbb{R}^n,\text{standard }g)$, $\nabla$ is ordinary differentiation in standard coordinates; in the standard global orthonormal frame, the connection 1-form on $O(\mathbb{R}^n)$ is identically zero.
2. 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 $n\ge 2$.
3. 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 $\nabla_XY=\tfrac12[X,Y]$.