Representation for a Smooth Isometric Mapping from a Connected Planar Domain to a Surface

  • Yi-Chao Chen
  • Roger Fosdick
  • Eliot FriedEmail author


A representation theorem for a smooth isometric mapping of a flat, connected domain \(\mathcal {D}\) in two-dimensional Euclidean point space \(\mathcal {E}^{2}\) into a surface \(\mathcal {S}\) in three-dimensional Euclidean point space \(\mathcal {E}^{3}\) is presented. The form of the mapping is shown to be necessary and sufficient to describe any such smooth isometry. Importantly, this work is not based upon the hypothesis that the mapped surface is ruled. In general, a mapping from a flat planar domain into a ruled surface is far from being isometric, and the property of being ruled is a partial consequence of our representation theorem.


Isometric embedding Rigidity Representation theorem 2-d to 3-d 

Mathematics Subject Classification

53A05 74K15 74K35 57R40 53A45 


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of HoustonHoustonUSA
  2. 2.Department of Aerospace Engineering and MechanicsUniversity of MinnesotaMinneapolisUSA
  3. 3.Mathematical Soft Matter UnitOkinawa Institute of Science and Technology Graduate UniversityOnnaJapan

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