TFAE: Metric-compatible connections on a metric vector bundle

Let $M$ be a smooth manifold and let $(E,\langle\cdot,\cdot\rangle)\to M$ be a real vector bundle of rank $r$ equipped with a smoothly varying inner product on fibers. Let $\nabla$ be a connection on a vector bundle on $E$ .

The following are equivalent:

1. Metric preservation (Leibniz rule for the inner product). For all smooth vector fields $X$ on $M$ and smooth sections $s,t$ of $E$ ,

X\langle s,t\rangle \;=\; \langle \nabla_X s, t\rangle \;+\; \langle s,\nabla_X t\rangle.

2. Vanishing covariant derivative of the metric. The covariant derivative $\nabla\langle\cdot,\cdot\rangle$ (viewed as a tensor) is identically zero; equivalently, the connection induced by $\nabla$ on $E^*\otimes E^*$ annihilates the section representing the metric.

3. Skew connection $1$ -forms in orthonormal frames. On any open set $U$ with a local orthonormal frame $(e_1,\dots,e_r)$ , the connection is described by matrix-valued $1$ -forms $\omega=(\omega^i{}_j)$ via

\nabla e_j = \sum_i \omega^i{}_j\, e_i,

and $\omega$ takes values in $\mathfrak{so}(r)$ , i.e. $\omega^i{}_j+\omega^j{}_i=0$ .

4. Isometric parallel transport. For every smooth curve $\gamma:[0,1]\to M$ , the parallel transport map on fibers is an isometry:

\langle \mathrm{PT}_\gamma(v), \mathrm{PT}_\gamma(w)\rangle \;=\; \langle v,w\rangle \quad\text{for all }v,w\in E_{\gamma(0)},

where $\mathrm{PT}_\gamma$ denotes parallel transport determined by $\nabla$ .

5. Orthonormal frame bundle reduction. The bundle of orthonormal frames $O(E)\to M$ is a principal G-bundle with structure group $\mathrm{O}(r)$ , and the connection induced by $\nabla$ on the full frame bundle restricts to a principal $\mathrm{O}(r)$ -connection on $O(E)$ .

Holonomy and the specified metric

These conditions imply that holonomy preserves the given metric in each fiber. The converse statement based only on holonomy at one point requires care: on a connected base, an inner product preserved by holonomy at that point extends by parallel transport to a parallel bundle metric, which need not equal a previously specified metric on the whole bundle. For example, the trivial connection on the real line bundle over $\mathbb R$ has trivial holonomy, but does not preserve the metric $h_x(v,w)=e^{2x}vw$ .

Examples

Levi-Civita connection on the tangent bundle. On a Riemannian manifold, the Levi-Civita connection on the tangent bundle is metric-compatible by definition; its parallel transport preserves the Riemannian inner product on tangent spaces.

Trivial bundle with constant metric and trivial connection. If $E=M\times\mathbb{R}^r$ carries the standard dot product fiberwise and $\nabla$ is the componentwise derivative in the trivialization, then the connection forms are zero (hence skew), and parallel transport is the identity, so $\nabla$ is metric-compatible.

Matrix-valued connection with values in so(r). On a trivial rank- $r$ bundle, define $\nabla=d+A$ where $A$ is an $\mathfrak{so}(r)$ -valued $1$ -form. Then the connection preserves the standard metric and has holonomy contained in $\mathrm{O}(r)$ ; nonzero curvature can occur even though metric compatibility holds.