Let $S=[0,1]^2$. Ignoring issues having to do with boundaries and corners, a chart is a diffeomorphism $\varphi \colon S \to S$. Let $g_0$ denote the flat (euclidean) metric. Given a chart $\varphi \colon (S, g_0) \to (S, g_0)$, which need not be an isometry, consider the curves $\alpha_u(t)=\varphi^{-1}(u,t)$ and $\beta_v(t)=\varphi^{-1}(t,v)$ in $S$ which are assumed to not be geodesics.

Does there exist a harmonic map $F: (S,g_0)\to T^2 $ for which $F(\alpha_u(t))$ and $F(\beta_v(t))$ are all geodesics?

Here $T^2$ has the quotient metric of $\mathbb R^2\backslash \mathbb Z^2$.

One way to view this is to only assume a local diffeo (which is not really my question) I tink is to consider torsion free connections implying projectively equivalent manifolds (can re-parametrize geodesics on one or the other to get same unparametrized geodesics).

I think the main question is whether this is true:

$$\mathfrak{p_0}\in \mathfrak{P}_H(M).$$

with $\mathfrak{p}_0:=\varphi^*[\nabla^{g_0}]=[\nabla^{\varphi^* g_0}],$ where we have

$$\mathfrak{P}_{LC}(M)=\{[\nabla^g]:g \text{ Riemannian metric}\} $$

and here $LC$ is the Levi-Civita connection.

Recall that given a local diffeomorphism $F\colon M\to N$ between manifolds, pulling back connections induces an isomorphism $F^*\colon \mathfrak{P}(N)\to \mathfrak{P}(M).$

Suppose now that $(M,g)$ and $(N,h)$ are Riemannian, then we have the following subset of $\mathfrak{P}_{LC}(M):$ $$ \mathfrak{P}_H(M) = \{F^*[\nabla^h]: F\colon (M,g)\to(N,h) \text{ harmonic local diffeo}\} \subset \mathfrak{P}_{LC}(M) $$

So it would be good to understand how to characterize $\mathfrak {P}_H(M)$ and know if $\mathfrak {p_0} \in \mathfrak{P}_H(M).$

How do we finish this? Does there exist a harmonic map $F: (S,g_0)\to T^2 $ for which $F(\alpha_u(t))$ and $F(\beta_v(t))$ are all geodesics?