The origins of the fundamental theorem of surface theory
The paper traces the history of two related results in classical differential geometry, namely the Mainardi-Codazzi equations, which provide relations between the first and the second fundamental form, and the fundamental theorem of surface theory, which guarantees the existence and uniqueness (up to rigid motions) of a surface with prescribed first and second fundamental form. Both results are closely linked to the Grand Prix des Mathématiques announced by the Académie des Sciences de Paris in 1859, which focused on the problems of applicability of surfaces (where the term ``applicable'' is understood as ``locally isometric''). The present paper provides a detailed overview and comparison between the prize memoirs of Edmond Bour (who received the prize), Ossian Bonnet and Delfino Codazzi (who both received an honorable mention). It also traces some subsequent developments, in particular Bonnet's 1867 proof of the fundamental theorem.
scientific article; zbMATH DE number 7629767