Note Note 27 — § 5 EINSTEIN'S THEORY OF GRAVITATION

(p. 53). The theory of surfaces, i.e. the study of geometry upon surfaces, makes it immediately apparent that the theorems, which have been established for any surface, also hold for any surface which can be generated by distorting the first without tearing. For if two surfaces have a point-to-point correspondence, such that the line-elements are equal at corresponding points, then corresponding finite arcs, angles, and areas, etc., will be equal. One thus arrives at the same planimetrical theorems for the two surfaces. Such surfaces are called "deformable" surfaces. The necessary and sufficient condition that surfaces be continuously deformable is that the expression for the line-element of the one surface can be transformed into that for the other, According to Gauss, it is necessary that both surfaces have equal measures of curvature. If the latter is constant over the whole surface, as e.g. in the case of a cylinder or a plane, all conditions for the deformability of the surfaces are fulfilled. In other cases, special equations offer a criterion as to whether surfaces, or portions of surfaces, are deformable into one another. The numerous subsidiary problems, which result out of these questions, are discussed at length in every book dealing with differential geometry (e.g. Bianchi-Lukat).[17] This branch of training, which was hitherto of interest only to mathematicians, now assumes very considerable importance for the physicist too.