If we know the Gaussian curvature and/or mean curvature of a surface embedded in R3, is it possible to reconstruct the original surface? If yes, how would one go about doing such a thing?
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See also Does the curvature determine the metric? – mr_e_man Jan 31 '22 at 18:09
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@zwim What do curves have to do with surfaces? – Ted Shifrin Jan 31 '22 at 19:43
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I just didn't pay attention the linked post was for curves, that'all. – zwim Feb 01 '22 at 09:44
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According to the following question posted previously and already answered is not possible: Gaussian curvature and mean curvature sufficient to characterize a surface?. According to the Bonnet theorem (https://encyclopediaofmath.org/wiki/Bonnet_theorem).
Andrew M.
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1I think you misunderstood the linked answer: The surface is unique up to a rigid motion of ${\mathbb R}^3$. – Moishe Kohan Jan 31 '22 at 18:04
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If you know both curvatures, the surface is unique up to rigid motion. If you know only one, there is no uniqueness. – Hans Engler Jan 31 '22 at 18:31
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More precisely, once you prescribe the curvature and 2nd fundamental form, not the mean curvature, then the surface is determined. – Moishe Kohan Jan 31 '22 at 18:32
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There's also a whole issue of integrability. I cannot prescribe arbitrary first and second fundamental forms and expect to get even a piece of a surface; the Gauss and Mainardi-Codazzi equations are necessary and sufficient conditions to be able to integrate. I see now that the linked answer already tells you that. – Ted Shifrin Jan 31 '22 at 19:45