Imagine a metric surface M (possibly noncompact, and possibly with boundary) which, in the complement of a discrete set C = C(M), is a flat Riemannian surface. We hold off on insisting on a particular differentiability class.
We call C(M) the "conepoints" of M.
At each point p of D, however, p has a neighborhood which is isometric to a cone (that is, one nappe of a cone) having (total angle about p) = ang(p), with ang(p) unequal to 2 pi.
(These bear a resemblance to orbifolds, but be that as it may.)
Let's tentatively agree to call such things FCP-surfaces. (Note that FCP-surfaces are defined abstractly without reference to an ambient space.)
For example, any compact polyhedron in R^3 has an underlying FCP-surface.
(Each FCP-surface may be considered to be the limit of a sequence of genuine Riemannian surfaces. It's easy to show that for compact FCP-surfaces without boundary, there is (unsurprisingly) a Gauss-Bonnet theorem:
sum defect(p) = 2 pi X(M) p in D
where defect(p) = 2 pi - ang(p), and X = Euler characteristic.
There is also a version of this for compact M with boundary.)
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QUESTIONS: Suppose M is an FCP-surface which topologically embeds (immerses) in R^3. Under what conditions does it do so isometrically??? And when can the embedding (immersion) be smooth in the complement of the conepoints?
(E.g., an FCP-surface M arising from a compact convex polyhedron in R^3 probably cannot be isometrically embedded smoothly in the complement of C(M).)
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For starters, what about FCP S^2's ??? Or FCP planes ???
Any reference to the literature will also be appreciated.
Dr. Daniel Asimov Senior Research Scientist
Mail Stop T27A-1 NASA Ames Research Center / MRJ Moffett Field, CA 94035-1000
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