A connection on a (pseudo) 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 \({O(r,s)}\), or of \({SO(r,s)}\) if (and only if) \({M}\) is orientable.

In terms of the metric, this can be written \({g_{ab}\parallel_{C}v^{a}\parallel_{C}w^{b}=g_{ab}v^{a}w^{b}}\). But recalling that the parallel transport of tensors just transports the arguments, we also have \({\left(\parallel_{-C}g_{ab}\right)v^{a}w^{b}=g_{ab}\parallel_{C}v^{a}\parallel_{C}w^{b}}\), so that we must have \({\parallel_{-C}g_{ab}=g_{ab}}\), or \({\nabla_{c}g_{ab}=0}\). 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. In terms of the connection coefficients, a metric connection then satisfies

\begin{aligned}\nabla_{c}g_{ab} & =\partial_{c}g_{ab}-\Gamma^{d}{}_{ac}g_{db}-\Gamma^{d}{}_{bc}g_{ad}=0\\
\Rightarrow\partial_{c}g_{ab} & =\Gamma{}_{abc}+\Gamma{}_{bac},
\end{aligned}

where we write \({\Gamma{}_{abc}\equiv\Gamma^{d}{}_{bc}g_{ad}}\), which again it is important to note is not tensor. By considering \({\partial_{c}\left(g^{ad}g_{df}\right)=\partial_{c}\left(\delta^{a}{}_{f}\right)=0}\), we arrive at the complementary expression

\begin{aligned}\partial_{c}g^{ab} & =-g^{ad}g^{bf}\partial_{c}g_{df}\\
& =-\left(\Gamma^{ab}{}_{c}+\Gamma^{ba}{}_{c}\right).
\end{aligned}

The Levi-Civita connection (AKA Riemannian connection, Christoffel connection) is then the torsion-free metric connection on a (pseudo) Riemannian manifold \({M}\). It can be shown that when \({M}\) is isometrically embedded in some \({\mathbb{R}^{n}}\), the Levi-Civita connection is the tangential connection induced by that of \({\mathbb{R}^{n}}\) according to this embedding. Furthermore, the autoparallel geodesics defined by the Levi-Civita connection can be shown to be identical to the Riemannian geodesics defined by the metric, so that “straight” and “extremal distance” paths coincide. We will denote the Levi-Civita connection and related quantities with an overbar, e.g. \({\overline{\Gamma}}\), \({\overline{\nabla}}\), and \({\overline{R}}\).

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 connected manifold \({M}\) with a torsion-free connection, a metric of signature \({(r,s)}\) compatible with this connection exists if and only if \({\mathrm{Hol}(M)\subseteq O(r,s)}\); in fact, any metric which is invariant under \({\mathrm{Hol}(M)}\) is such a compatible metric.

In the case of a general connection where \({\mathrm{Hol}(M)=SO(n)}\) on an orientable \({M}\), the associated metric is unique up to a scaling factor (in physics, this corresponds to a choice of units). If \({\mathrm{Hol}(M)=SO(n-1,1)}\), then there is a separate scaling factor for the time-like component. If \({\mathrm{Hol}(M)}\) is the identity, then the connection is flat, and the metric is determined by a choice of inner product (of any signature) at any point, which applies to all other points via the unique parallel transport.

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 for zero torsion, this leads to another commonly noted symmetry

\(\displaystyle \overline{R}_{cdab}=\overline{R}{}_{abcd}. \)

The Leibniz rule for the covariant derivative over the inner product along with the zero torsion relation \({\overline{\nabla}_{v}w=\overline{\nabla}_{w}v+\left[v,w\right]}\) can be used to derive an expression called the Koszul formula:

\(\displaystyle \begin{aligned}2\left\langle \overline{\nabla}_{u}v,w\right\rangle =&\overline{\nabla}_{u}\left\langle v,w\right\rangle +\overline{\nabla}_{v}\left\langle w,u\right\rangle -\overline{\nabla}_{w}\left\langle u,v\right\rangle \\&-\left\langle u,[v,w]\right\rangle +\left\langle v,[w,u]\right\rangle +\left\langle w,[u,v]\right\rangle \end{aligned} \)

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

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

On a (pseudo) Riemannian manifold, the connection coefficients for the Levi-Civita connection in a coordinate basis \({\overline{\Gamma}^{\lambda}{}_{\mu\sigma}}\) are called the Christoffel symbols, and are sometimes denoted \({\{\substack{\lambda\\ \mu\sigma}\}}\) or \({\{\substack{\mu\sigma\\ \lambda}\}}\). At a point \({p\in U\subset M}\), an orthonormal basis for \({T_{p}U}\) can be used to form geodesic normal coordinates, which are then called Riemann normal coordinates. Recalling that with zero torsion the connection coefficients vanish at \({p}\), we can apply the covariant derivative to the metric tensor to conclude that the partial derivatives of the metric \({g_{\mu\nu}=\eta_{\mu\nu}}\) all also vanish at \({p}\).