A surface is said to be applicable (or locally isometric) to another one if for all the points on the first surface, there exists a one-to-one application from a neighborhood of this point (in the surface) onto a part of the second one, such that the geodesic distance is preserved.

If parametrizations of both the surfaces with equal second fundamental form can be found, then they are applicable (see the notations).

Examples:
     - the surfaces of class C² applicable to a plane are the developable surfaces (therefore, they are ruled surfaces). But, by connecting bits of cones and planes, surfaces applicable to a plane and that are not ruled can be constructed (like also a crumpled piece of paper !).

     - the surfaces with given constant total curvature are applicable (and, in particular, are applicable to a sphere if the curvature is positive, and to a pseudo-sphere if it is negative).

     - All helicoids are applicable to a surface of revolution (Bour theorem); in particular, the right helicoid is applicable to the catenoid.

The "theorema egregium" ("remarkable theorem") of Gauss states that two applicable surfaces have the same total curvature at corresponding points.
Its converse is false (cf [Audin], p. 270).
 
 

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© Robert FERRÉOL 2017