this problem is ill-posed; however, generalized solutions exist if a finite number of singular points are removed from $M$ [@viertel_approach_2019; @bethuel_ginzburg-landau_1994].

This strategy can be extended to surfaces by replacing the gradient operator $\nabla$ with the appropriate *connection* on the tangent bundle. The Levi-Civita connection provides a way to compare vectors on the tangent bundle that preserves the notion of inner product between tangent spaces. That is, if $P_{pq}$ is the parallel transport function for the Levi-Civita connection between $T_p$ and $T_q$, and if ${v}_1, {v}_2 \in T_p$, then $$\langle{v}_1,{v}_2 \rangle_{T_P} = \langle P_{pq}({v}_1),P_{pq}({v}_2) \rangle_{T_q}.$$

Visually, the effect of using this connection in equation \[eq:energy\] is that a minimizing vector field appears smooth (see ).

![image](fields.pdf){width="\textwidth"}

In order to extend the strategy from 2D, using equation \[eq:energy\] to design a smooth cross field, we seek a connection, $\nabla^Q$, on the tangent bundle with corresponding parallel transport function, $Q$, appropriate for transporting *representation vectors*. We choose $Q$ in such a way that the component vectors of the corresponding crosses are transported by $P$, the parallel transport function corresponding to the Levi-Civita connection. Let $\gamma_{pq}(t) \colon [0,1] \rightarrow M$ be a Levi-Civita geodesic connecting points $p$ and $q$ such that $\gamma_{pq}(t_p) = 