Results in Mathematics

, Volume 27, Issue 1–2, pp 97–104 | Cite as

An intrinsic characterization of developable surfaces

  • Barbara Opozda


We consider the classical theorem saying that if f: M → R3 is a Riemannian surface in R3 without planar points and with vanishing Gaussian curvature, then there is an open dense subset M′ of M such that around each point of M′ the surface f is a cylinder or a cone or a tangential developable. As we shall show below, the theorem, in fact, belongs to affine geometry. We give an affine proof of this theorem. The proof works in Riemannian geometry as well. We use the proof for solving the realization problem for a certain class of affine connections on 2-dimensional manifolds. In contrast with Riemannian geometry, in affine geometry, cylinders, cones as well as tangential developables can be characterized intrinsically, i.e. by means of properties of any nowhere flat induced connection. According to the characterization we distinguish three classes of affine connections on 2-dimensional manifolds, i.e. cylindric, conic and TD-connections.


type number cylinders cones tangential developables 

1990 MS Classification

53A05 53B05 


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Copyright information

© Birkhäuser Verlag, Basel 1995

Authors and Affiliations

  • Barbara Opozda
    • 1
  1. 1.Instytut Matematyki UJ ul.KrakówPoland

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