Results of the study of Abelian groups of hypersurfaces (i.e., hypersurfaces that form the cross sections of a one parameter group of point transformations in an n-dimensional metric space). Part one obtains the basic defining differential equations of the Abelian group of hypersurfaces; part two examines critical points in the enveloping n-dimensional space that belong to two or more distinct hypersurfaces of the group; and part three obtains the images of the first and second fundamental forms and the field of normal vectors of a given hypersurface when it is imbedded in an Abelian group of hypersurfaces.
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