The Levi-Civita connection

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},

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).

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 \({p}\).

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.

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