The Riemann-Christoffel tensor is derived solely from the~$g_{\mu\nu}$ and therefore belongs to the class of fundamental tensors. Usually we can form from any tensor a series of tensors of continually increasing rank by covariant \PageSep{73} differentiation. But this process is frustrated in the case of the fundamental tensors because $g_{\mu\nu\sigma}$~vanishes identically. We have got round the gap and reached a fundamental tensor of the fourth rank. The series can now be continued indefinitely by covariant differentiation. When the Riemann-Christoffel tensor vanishes, the differential equations \index{Integrability of parallel displacement}% \index{Riemann-Christoffel tensor!vanishing of}% \[ A_{\mu\nu} = \frac{\dd A_{\mu}}{\dd x_{\nu}} - \{\mu\nu, \alpha\}\, A_{\alpha} = 0 \Tag{(34.7)} \] are integrable. For the integration will be possible if \Eq{(34.7)} makes~$dA_{\mu}$ or \[ \frac{\dd A_{\mu}}{\dd x_{\nu}}\, dx_{\nu} \] a complete differential, i.e.\ if \[ \{\mu\nu, \alpha\}\, A_{\alpha}\, dx_{\nu} \] is a complete differential. By the usual theory the condition for this is \[ \frac{\dd}{\dd x_{\sigma}} (\{\mu\nu, \alpha\}\, A_{\alpha}) - \frac{\dd}{\dd x_{\nu}} (\{\mu\sigma, \alpha\}\, A_{\alpha}) = 0, \] or \[ A_{\alpha} \left(\frac{\dd}{\dd x_{\sigma}} \{\mu\nu, \alpha\} - \frac{\dd}{\dd x_{\nu}} \{\mu\sigma, \alpha\}\right) + \{\mu\nu, \alpha\}\, \frac{\dd A_{\alpha}}{\dd x_{\sigma}} - \{\mu\sigma, \alpha\}\, \frac{\dd A_{\alpha}}{\dd x_{\nu}} = 0. \] Substituting for $\dd A_{\alpha}/\dd x_{\sigma}$, $\dd A_{\alpha}/dx_{\nu}$ from~\Eq{(34.7)} %[** TN: Not broken in the original] \begin{multline*} A_{\alpha} \left(\frac{\dd}{\dd x_{\sigma}} \{\mu\nu, \alpha\} - \frac{\dd}{\dd x_{\nu}} \{\mu\sigma, \alpha\}\right) \\ + (\{\mu\nu, \alpha\}\, \{\alpha\sigma, \epsilon\} - \{\mu\sigma, \alpha\}\, \{\alpha\nu, \epsilon\}) A_{\epsilon} = 0. \end{multline*} Changing the suffix~$\alpha$ to~$\epsilon$ in the first term, the condition becomes \[ A_{\epsilon} B_{\mu\sigma\nu}^{\epsilon} = 0. \] Accordingly when $B_{\mu\sigma\nu}^{\epsilon}$~vanishes, the differential~$dA_{\mu}$ determined by~\Eq{(34.7)} will be a complete differential, and \[ \int dA_{\mu} \] between any two points will be independent of the path of integration. We can then carry the vector~$A_{\mu}$. by parallel displacement to any point obtaining a unique result independent of the route of transfer. If a vector is displaced in this way all over the field, we obtain a \emph{uniform vector-field}. \index{Uniform!vector-field}% This construction of a uniform vector-field is only possible when the Riemann-Christoffel tensor vanishes throughout. In other cases the equations have no complete integral, and can only be integrated along a particular route. E.g., we can prescribe a \emph{uniform direction} at all points of a plane, but their is nothing analogous to a uniform direction over the surface of a sphere.