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If I start with an infinite flat sheet of graph paper, and in polar coordinates cut out a piece according to: $r>0, \ \ -f(r) < \theta < f(r)$

Now I want to stitch the remaining graph paper together, by associating each point $(r,f(r))$ to $(r,-f(r))$ on the seam.

How do I calculate what the curvature is of the stitched up paper along the seam?

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    Are you sure it is possible to stitch the paper together?2011-04-30
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    I don't see why not. The top and bottom seam is a mirror image across $\theta=0$, so I could easily stitch these in real life. So I don't see any problem doing it theoretically. Is there some requirement I should know about for stitching seams together? Is it more than assigning points on the seam smoothly to each other?2011-04-30
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    Unless I am misunderstanding, one has the following problem: If $f(r)$ is not constant (i.e. if the top and bottom seams are not straight) then in real life you would have to essentially arrange the paper so that near the seams the top and bottom of the paper were parallel (so that you match the two seams). The sewn piece of paper will then have a cusp all the way along the seam (so the curvature will be infinite along the seam). If you try to do it any other way, you will tear the paper while trying to sew up the seams. (But if the seam is a straight line, then you can match the two ...2011-04-30
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    ... edges of the paper flatly along the seam, and then have curvature zero along the seam --- all the curvature will be concentrated at the cone point.)2011-04-30
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    So the curvature is either zero or infinite (a delta function like limit)? And the only way to get zero is with a straight line? This doesn't seem intuitive to me. Can you show the math in an answer so I can learn the details? That would be great!2011-04-30
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    @John In a sufficiently small neighbourhood of any point not on your seam, the (Gaussian) curvature is zero, since it is isometric to the plane. So all the curvature has to be concentrated in the seam; a set of measure zero. The integral of the curvature in a small disk that intersects the seam cannot be nonzero if the curvature on the seam is represented by finite values.2011-04-30
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    @yasmar Okay, the only possibilities being zero or infinite curvature at a point on the seam makes sense now. Thank you. That the only way to get zero curvature is with f(r)=constant still doesn't though. Maybe just an answer showing how I could calculate the curvature at the seam would be a great start.2011-04-30
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    Take a look at [this answer by Deane Yang](http://mathoverflow.net/questions/16850/curvature-and-parallel-transport/17099#17099) at MathOverflow to get an idea of how curvature can be computer via parallel transport of a vector around a closed loop. On a two-dimensional surface this computation simplifies drastically (see, e.g. exercise 12 in [this problem set](http://www.dpmms.cam.ac.uk/study/II/DifferentialGeometry/2010-2011/dg2e4.pdf); note that the computation is still morally valid if your manifold is only $C^1$, as it would be in your case).2011-04-30

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