Is the inverse stereographic projection the exponential map of the South pole?
When I say "inverse stereographic projection" I mean the projection from the plane tangent to the South pole to the sphere, i.e. the opposite of this picture from the corresponding Wolfram article.
As far as I understand, exponential maps allow one to map lines in the tangent plane of a point (here the distinguished point is the South pole) to geodesics in an open neighborhood around the point on the manifold (which in this case would be $S^2 \setminus N$, where $N$ denotes the north pole).
It does seem like the stereographic projection maps line segments through the origin to arcs, which are the geodesics of the sphere, containing the South pole.
I think this might be similar to what the second example of an exponential map on p.232, chapter 5, of Kuehnel's Differential Geometry, is saying, although I am not positive because it is written using polar and spherical coordinates -- however the example is supposed to degenerate at the North pole, just like the stereographic projection would. On p. 55 of these lecture notes it states that stereographic projection induces a correspondence between geodesics, as I thought it did. The third and last page of this document seems to say that the Mercator projection followed by the exponential map gives the same result as the stereographic projection.
The only reason why the Stereographic projection seemingly wouldn't be an exponential map is because it is very clearly not isometric. The fact that one has to first apply the Mercator projection before applying the (complex) exponential map (and the Mercator projection isn't even conformal like the Lambert projection) in order to get the stereographic projection also suggests that the latter cannot be isometric. However, I am not certain how accurately I am understanding this.
No; the exponential map at the south pole sends a circle of radius $\pi$ to the north pole (and generally wraps annuli $k\pi < r < (k + 1)\pi$ over the complement of the two poles), while stereographic projection is a diffeomorphism to its image (the complement of the north pole).