Characterizations of Metric Connections

Theorem

Let $\struct {M, g}$ be a Riemannian or pseudo-Riemannian manifold (with or without boundary).

Let $TM$ be the tangent bundle of $TM$.

Let $\nabla$ be a connection on $TM$.

Let $\map {\mathfrak X} M$ be the space of smooth vector fields on $M$.

Let $\gamma$ be a smooth curve in $M$.

For all $i \in \N_{>0} : i \le \dim M$ let $\tuple {E_i}$ be a smooth local frame for $TM$.

Let $\set {\Gamma^k_{ij} }$ be the connection coefficients with respect to $\tuple {E_i}$.

Let $D_t$ be the covariant derivative along $\gamma$.

Then the following properties are equivalent:

Property $1$

$\forall X, Y, Z \in \map {\mathfrak X} M : \nabla_X \innerprod Y Z = \innerprod {\nabla_X Y} Z + \innerprod Y {\nabla_X Z}$

Property $2$

$\nabla g = 0$

Property $3$

$\Gamma^l_{ki} g_{lj} + \Gamma^l_{kj} g_{il} = \map {E_k} {g_{ij} }$

Property $4$

$\dfrac \d {\d t} \innerprod Y Z = \innerprod {D_t Y} Z + \innerprod Y {D_t Z}$

Property $5$

If $V, W \in \map {\mathfrak X} \gamma$, then $\innerprod V W$ is constant along $\gamma$

Property $6$

Parallel transport map along $\gamma$ is a linear isometry

Property $7$

Orthonormal basis at a point of $\gamma$ can be extended to a parallel orthonormal frame along $\gamma$

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Connections on Abstract Riemannian Manifolds