We now impose two additional conditions on the connection, then argue that these are enough to determine the connection uniquely.

First, we require that \begin{equation} d(\vv\cdot\ww) = d\vv\cdot\ww + \vv\cdot d\ww \end{equation} so that differentiation respects the dot product. Applying this condition to our orthonormal bases, we have \begin{equation} 0 = d(\ee_i\cdot\ee_j) = d\ee_i\cdot\ee_j + \ee_i\cdot d\ee_j = \omega_{ji} + \omega_{ij} \label{mcomp} \end{equation} A connection which satisfies ($\ref{mcomp}$) is called metric compatible.

Next, we consider $d^2$ on vectors. We would like to demand that this be zero, but that turns out to be too strong. Consider for example $d\rr$. In the standard Euclidean spaces of vector calculus, it is clear that \begin{equation} d\rr = d(\rr) \end{equation} where $\rr$ is the position vector, and it is then reasonable to expect that \hbox{$d^2\rr=0$}. But on the surface of a sphere, for example, $d\rr$ makes perfect sense, but $\rr$ doesn't exist! In this context, $d\rr$ is not “$d$” of anything, so it is not obvious what $d^2\rr$ should be.

We will nonetheless assume that \begin{equation} d^2\rr = 0 \label{d2rvec} \end{equation} but we will be careful not to assume that $d^2\vv$ is zero for other vectors — which, as we will see shortly, turns out to be false. Expanding (\ref{d2rvec}), we have \begin{equation} 0 = d^2\rr = d(d\rr) = d\left(\sigma^j\,\ee_j \right) = d\sigma^j\,\ee_j - \sigma^j\wedge d\ee_j \end{equation} so that \begin{equation} 0 = \ee_k \cdot d^2\rr = g_{kj} \,d\sigma^j - \sigma^j\wedge\omega_{kj} \end{equation} Rearranging terms and relabeling some indices, we have \begin{equation} 0 = g_{ki} \,d\sigma^i + \omega_{kj}\wedge\sigma^j = g_{ki} \left( d\sigma^i + \omega^i{}_j\wedge\sigma^j \right) \end{equation} Since the matrix $(g_{ij})$ is invertible, we can conclude that \begin{equation} 0 = d\sigma^i + \omega^i{}_j\wedge\sigma^j \label{tfree} \end{equation} A connection satisfying ($\ref{tfree}$) is called torsion free, and a connection which is both torsion free and metric compatible is called a Levi-Civita connection.

We claim that, given $d\rr$, there is a unique Levi-Civita connection. For now, we give only a rough justification of this fact, by counting the degrees of freedom. In $n$ dimensions, a connection is determined by specifying the $n^2$ 1-forms $\omega_{ij}$ in any basis. Metric compatibility ($\ref{mcomp}$) forces the $n$ “diagonal”connection 1-forms $\omega_{ii}$ to be zero, and relates the remaining connection 1-forms in pairs, reducing the number of independent connection 1-forms to $n(n-1)/2$, each with $n$ components. But the torsion free condition consists of $n$ 2-form equations, each with $n(n-1)/2$ components. Thus, the number of linear equations exactly matches the number of degrees of freedom, and we expect a unique solution. 1) A more complete derivation is given in §Uniqueness of the Levi-Civita Connection.