Development (differential geometry) In classical differential geometry , development refers to the simple idea of rolling one smooth surface over another in Euclidean space . For example, the tangent plane to a surface at a point can be rolled around the surface to obtain the tangent plane at other points.Properties The tangential contact between the surfaces being rolled over one another provides a relation between points on the two surfaces. If this relation is a bijection between the surfaces, then the two surfaces are said to be developable on each other or developments of each other. Differently put, the correspondence provides an isometry , locally, between the two surfaces. In particular, if one of the surfaces is a plane, then the other is called a developable surface: thus a developable surface is one which is locally isometric to a plane. The cylinder is developable, but the sphere is not.Development can be generalized further using flat connections . From this point of view , rolling the tangent plane over a surface defines an affine connection on the surface, and a developable surface is one for which this connection is flat. More generally any flat Cartan connection on a manifold defines a development of that manifold onto the model space . Perhaps the most famous example is the development of conformally flat n -manifolds, in which the model-space is the n -sphere. The development of a conformally flat manifold is a conformal local diffeomorphism from the universal cover of the manifold to the n -sphere.Undevelopable surfaces The class of double-curved surfaces contains objects that cannot be simply unfolded. Such surfaces can be developed only approximately with some distortions of linear surface elements
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