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Embedding FlatSurfaceswithConePoints in 3Space
Posted:
Jun 25, 1996 8:36 PM


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 FCPsurfaces. (Note that FCPsurfaces are defined abstractly without reference to an ambient space.)
For example, any compact polyhedron in R^3 has an underlying FCPsurface.
(Each FCPsurface may be considered to be the limit of a sequence of genuine Riemannian surfaces. It's easy to show that for compact FCPsurfaces without boundary, there is (unsurprisingly) a GaussBonnet 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.)
* * *
QUESTIONS: Suppose M is an FCPsurface 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 FCPsurface M arising from a compact convex polyhedron in R^3 probably cannot be isometrically embedded smoothly in the complement of C(M).)
* * *
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 T27A1 NASA Ames Research Center / MRJ Moffett Field, CA 940351000
asimov@nas.nasa.gov (415) 6044799 w (415) 6043957 fax



