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Abstract:
Geometry is topology plus extra structure—notions like angle,
distance, and curvature. In the case of surfaces, you might ask
about all the different ways to metrize a given topological
surface. If you were really demanding, you might ask for a way of
measuring how different two metrics are from each other: you'd be
asking for a metric on the space of metrics. It turns out that
this is a reasonable demand to make, and that one good answer comes
from so-called flat structures—a generalization of billiard
tables.
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