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Topic: Embedding Flat-Surfaces-with-Cone-Points in 3-Space
Replies: 4   Last Post: Jul 1, 1996 7:13 PM

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Daniel A. Asimov

Posts: 101
Registered: 12/3/04
Embedding Flat-Surfaces-with-Cone-Points in 3-Space
Posted: Jun 25, 1996 8:36 PM
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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.)

* * *

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).)

* * *

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
(415) 604-4799 w
(415) 604-3957 fax

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