An intrinsic characterization of developable surfaces
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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.
Keywordstype number cylinders cones tangential developables
1990 MS Classification53A05 53B05
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