What is the best way to motivate and explain what differential geometry to an audience of high school students? Any tips and suggestions are welcomed!
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I always thought "flat earth" vs. "spherical earth" seemed like a nice example to show that just because a space is locally Euclidean doesn't mean that whole space is Euclidean space.
Addendum: Also, the fact that everyone now knows the earth is spherical, but maps and atlases show locally planar representations is interesting for similar reasons.
I would describe the three different triangles you can make. The sum of the angles greater than, equal to, or less than 180 degrees. I suppose this would require you to talk about a geodesic, the shortest distance between to point in some space. From there I would motivate that with the paths that airplanes follow on earth, the great circles, which is why we fly over Greenland to get to Europe or over the Arctic from California to get to Russia. Then talk about what some aliens might do if they lived on a "planet" that happened to have negative curvature for the surface. Definitely try to use pictures, but the chalk board will have real limitations, unless you are a great drawer.
I suppose this only builds upon MPW suggestion, but Kids might like the comparison.