Projection of geodesics to a plane

Question

I was thinking if this statement holds true: For every geodesic on a surface there is a projection to a plane that sends it to a straight line.

Answer 1

The answer, suitably understood, is "yes, if $\nabla$ is nice, but only locally". An affine manifold $(M,\nabla)$ is called projectively flat if it can be covered by coordinate systems for which the $\nabla$-geodesics appear as straight lines. When $\nabla$ is torsionfree, we'll also say that $(M,\nabla)$ is locally equi-affine if its Ricci tensor is symmetric.

If $M$ is a surface, $\nabla$ is torsionfree, and $(M,\nabla)$ is locally equi-affine, then $(M,\nabla)$ is projectively flat if and only if the Codazzi equation $(\nabla_X{\rm Ric})(Y,Z) = (\nabla_Y{\rm Ric})(X,Z)$ holds, for all $X,Y,Z\in\mathfrak{X}(M)$. In particular, this holds whenever $\nabla$ is the Levi-Civita connection of a constant curvature metric (Riemannian or Lorentzian) on $M$.

A more general statement can be found on page 105 of:

H. Weyl; Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffassung, Göttingen Nachr. (1921), 99--112

Answer 2

This is true of one sheet hyperboloid asymptotic lines.

In all other cases it is true locally only... for tangents in the tangent plane, only at the point in consideration where geodesics intersect.