Without enough coffee in me, I can't seem to figure out how to post a comment. I don't think this qualifies as an answer, but that seems to be all I can figure out how to post. Hopefully MO will forgive me this breach.

I'm thinking that the trouble with the metric in the positively curved parts of the surface comes from the fact that when building these things in $R^3$ out of a polygon in a Euclidean or hyperbolic plane, we need to do a bit of stretching, because we run out of dimensions for just rolling - in $R^4$ a torus can be flat, yes? So how about if we fix a particular curvature for a curve on the surface, say the one corresponding to the shortest (in the embedded, Euclidean sense) generator of the fundamental group for a basepoint in the middle of the picture (again, in the embedded, Euclidean sense). Then we can scale tangents by the inverse of the curvature in the direction they specify, so that the directions that were the most stretched if we think of the surface as coming from a glued polygon are the easiest to move along. This could be done so that the osculating circles, once scaled, all end up the same size as our particular circle. This would have the effect, for example, of "tightening the belt" around the outside of the positively curved region, so that Deane Yang's flipping of the cylinder would happen.

I'm thinking that the trouble with the metric in the positively curved parts of the surface comes from the fact that when building these things in $R^3$ out of a polygon in a Euclidean or hyperbolic plane, we need to do a bit of stretching, because we run out of dimensions for just rolling - in $R^4$ a torus can be flat, yes? So how about if we fix a particular curvature for a curve on the surface, say the one corresponding to the shortest (in the embedded, Euclidean sense) generator of the fundamental group for a basepoint in the middle of the picture (again, in the embedded, Euclidean sense). Then we can scale tangents by the inverse of the curvature in the direction they specify, so that the directions that were the most stretched if we think of the surface as coming from a glued polygon are the easiest to move along. This could be done so that the osculating circles, once scaled, all end up the same size as our particular circle. This would have the effect, for example, of "tightening the belt" around the outside of the positively curved region, so that Deane Yang's flipping of the cylinder would happen.

I'm thinking that the trouble with the metric in the positively curved parts of the surface comes from the fact that when building these things in $R^3$ out of a polygon in a Euclidean or hyperbolic plane, we need to do a bit of stretching, because we run out of dimensions for just rolling - in $R^4$ a torus can be flat, yes? So how about if we fix a particular curvature for a curve on the surface, say the one corresponding to the shortest (in the embedded, Euclidean sense) generator of the fundamental group for a basepoint in the middle of the picture (again, in the embedded, Euclidean sense). Then we can scale tangents by the inverse of the curvature in the direction they specify, so that the directions that were the most stretched if we think of the surface as coming from a glued polygon are the easiest to move along. This could be done so that the osculating circles, once scaled, all end up the same size as our particular circle. This would have the effect, for example, of "tightening the belt" around the outside of the positively curved region, so that Deane Yang's flipping of the cylinder would happen.

Without enough coffee in me, I can't seem to figure out how to post a comment. I don't think this qualifies as an answer, but that seems to be all I can figure out how to post. Hopefully MO will forgive me this breach.

I'm thinking that the trouble with the metric in the positively curved parts of the surface comes from the fact that when building these things in $R^3$ out of a polygon in a Euclidean or hyperbolic plane, we need to do a bit of stretching, because we run out of dimensions for just rolling - in $R^4$ a torus can be flat, yes? So how about if we fix a particular curvature for a curve on the surface, say the one corresponding to the shortest (in the embedded, Euclidean sense) generator of the fundamental group for a basepoint in the middle of the picture (again, in the embedded, Euclidean sense). Then we can scale tangents by the inverse of the curvature in the direction they specify, so that the directions that were the most stretched if we think of the surface as coming from a glued polygon are the easiest to move along. This could be done so that the osculating circles, once scaled, all end up the same size as our particular circle. This would have the effect, for example, of "tightening the belt" around the outside of the positively curved region, so that Deane Yang's flipping of the cylinder would happen.