can you conclude from zero density points that an analytic function is zero in higher dimension? I know of this result for $f:\mathbb{R}\to\mathbb{R}$ but I am unsure if this holds for higher dimensions. In essence you should only get that the derivative is zero in one direction, the one where your approximating sequence is coming from. Ah wait, "density point" is different from the definition of "there exists an approximating sequence" I had in mind

So essentially these connections pull back straight lines in a particular chart and make them straight lines on the manifold (i.e. (m) connections pull back the chart mapping $\sum \mu_i \delta_i \mapsto (\mu_1,\dots,\mu_n)$ and (e) connections do the same for the dual of function spaces)? That seems a bit contrary to the entire premise of information geometry a la "Respect the space of distributions instead of using the geometry of an arbitrary parameter space of parametrized distributions"

What still confuses me, is that "straight lines" in my mind are supposed to encapsulate the notion of "shortest path". If the other connections result in geodesics ("straight lines"), which are not the shortest path, why are they reasonable definitions of "straight lines"? I.e. if $\nabla^{(e)}_{\dot{\gamma}(t)}(\dot{\gamma}(t)) = 0$ does not result in the shortest path, why is it reasonable?