The Levi-Civita connection

A connection on a Riemannian manifold $${M}$$ is called a metric connection (AKA metric compatible connection, isometric connection) if its associated parallel transport respects the metric, i.e. it preserves lengths and angles. More precisely, $${\forall v,w\in TM}$$, we require that $${\left\langle \parallel_{C}(v),\parallel_{C}(w)\right\rangle =\left\langle v,w\right\rangle }$$ for any curve $${C}$$ in $${M}$$. This means that the holonomy group is a subgroup of $${O(n)}$$, or of $${SO(n)}$$ if (and only if) $${M}$$ is orientable.

In terms of the metric, this can be written $${g_{ab}\parallel_{C}v^{a}\parallel_{C}w^{b}=g_{ab}v^{a}w^{b}}$$. Then since $${\nabla_{c}\left\langle v,w\right\rangle =0}$$, and recalling that the covariant derivative on the tensor algebra was defined to respect parallel translation, we also have for infinitesimal $${C}$$ that $${\parallel_{C}(g_{ab}v^{a}w^{b})=\parallel_{C}g_{ab}\parallel_{C}v^{a}\parallel_{C}w^{b}=g_{ab}v^{a}w^{b}}$$, so that we must have $${\parallel_{C}g_{ab}=g_{ab}}$$, or $${\nabla_{c}g_{ab}=0}$$. In terms of the connection coefficients, a metric connection then satisfies

$$\displaystyle \nabla_{c}g_{ab}=\partial_{c}g_{ab}-\Gamma^{d}{}_{ac}g_{db}-\Gamma^{d}{}_{bc}g_{ad}=0.$$

Using the Leibniz rule for the covariant derivative over the tensor product, we can derive a Leibniz rule over the inner product:

\displaystyle \begin{aligned}\nabla_{c}\left(g_{ab}v^{a}w^{b}\right) & =0+g_{ab}\nabla_{c}v^{a}w^{b}+g_{ab}v^{a}\nabla_{c}w^{b}\\ \Rightarrow\nabla_{u}\left\langle v,w\right\rangle & =\left\langle \nabla_{u}v,w\right\rangle +\left\langle v,\nabla_{u}w\right\rangle \end{aligned}

Requiring this relationship to hold is an equivalent way to define a metric connection.

The Levi-Civita connection (AKA Riemannian connection, Christoffel connection) is then the torsion-free metric connection on a (pseudo) Riemannian manifold $${M}$$. The fundamental theorem of Riemannian geometry states that for any (pseudo) Riemannian manifold the Levi-Civita connection exists and is unique. On the other hand, an arbitrary connection can only be the Levi-Civita connection for some metric if it is torsion-free and preserves lengths. More precisely, given a simply connected manifold $${M}$$ with a torsion-free connection, a metric of signature $${(r,s)}$$ compatible with this connection exists if and only if $${\mathrm{Hol}(M)\subseteq O(r,s)}$$; moreover, this metric is unique only up to a scaling factor (in physics, this corresponds to a choice of units).

For a metric connection, the curvature then must take values that are infinitesimal rotations, i.e. $${\check{R}}$$ is $${o(r,s)}$$-valued. Thus if we eliminate the influence of the signature by lowering the first index, the first two indices of the curvature tensor are anti-symmetric:

$$\displaystyle R{}_{cdab}=-R{}_{dcab}$$

Using the anti-symmetry of the other indices and the first Bianchi identity, this leads to another commonly noted symmetry

$$\displaystyle R{}_{cdab}=R{}_{abcd}.$$

The Leibniz rule for the covariant derivative over the inner product can be used to derive an expression called the Koszul formula:

\displaystyle \begin{aligned}2\left\langle \nabla_{u}v,w\right\rangle = & \nabla_{u}\left\langle v,w\right\rangle +\nabla_{v}\left\langle w,u\right\rangle -\nabla_{w}\left\langle u,v\right\rangle \\ & -\left\langle u,[v,w]\right\rangle +\left\langle v,[w,u]\right\rangle +\left\langle w,[u,v]\right\rangle \end{aligned}

Substituting in the frame vector fields and eliminating the metric tensor from the left hand side, we arrive at an expression for the connection in terms of the metric:

\displaystyle \begin{aligned}2\Gamma^{c}{}_{ba}=g^{cd}( & \partial_{a}g_{bd}+\partial_{b}g_{da}-\partial_{d}g_{ab}\\ & -g_{af}[e_{b},e_{d}]^{f}+g_{bf}[e_{d},e_{a}]^{f}+g_{df}[e_{a},e_{b}]^{f}) \end{aligned}

On a Riemannian manifold, the connection coefficients for the Levi-Civita connection in a coordinate basis $${\Gamma^{\lambda}{}_{\mu\sigma}}$$ are called the Christoffel symbols. Thus the Christoffel symbols are determined by the partial derivatives of the metric, which means that the Christoffel symbols vanish at the origin of Riemann normal coordinates.

 ◊ The vanishing of the Christoffel symbols at the origin of Riemann normal coordinates is frequently used to simplify the derivation of tensor relations which are then, being frame-independent, seen to be true in any coordinate system or frame (and if the origin was chosen arbitrarily, at any point). In particular, the covariant and partial derivatives are equivalent at the origin of Riemann normal coordinates.