... to find all surfaces of a given linear elementand this was exactly in the right area for Codazzi's research in differential geometry.
The 1859 Prize of the Academy of Sciences had three high quality entries. They were from the two Frenchmen Bour and Bonnet, and from the Italian Codazzi. All three pieces of work are important contributions to differential geometry but, although the manuscripts of Bour and Bonnet were published in Comptes-Rendus des séances de l'Académie des sciences fairly soon after the prize was awarded, Codazzi's entry was not published until 1883 (10 years after his death). Codazzi's submission for the prize contained a result which gave necessary and sufficient conditions for one surface to be mapped to another. It also contained his famous formulas which proved important in the theory of surfaces. These are now called the Mainardi-Codazzi formulas and we explain below the role played by Gaspare Mainardi in their discovery. The formulas give two relations between the first and second quadratic forms over a surface together with an equation, already found by Gauss, which gives necessary and sufficient conditions for the existence of a surface which admits two given quadratic forms. Bonnet used Codazzi's formulas to prove an existence theorem in the theory of surfaces.
Codazzi was not the first to discover these formulas since Mainardi, who taught at the University of Pavia, had published them in a paper of 1856. The two mathematicians, however, made their discoveries completely independently of each other. Although the two contributions were equivalent, Codazzi gave a simpler formulation, and much wider applications than Mainardi. The article  shows that, in 1853, Karl M Peterson, then a student of Minding at the University of Dorpat (now named Tartu), submitted a dissertation containing a derivation of two equations equivalent to those of Mainardi and Codazzi and outlining a proof of the fundamental theorem of surface theory.
Certainly Codazzi became well known for his contributions to differential geometry and this led to his appointment to a chair of algebra and analytic geometry at the University of Pavia in 1865. He was now in a much better position to extend his research with the security of a university post meaning he felt under far less pressure than he had before. His researches after his appointment were on curvilinear coordinates and he published these results in Sulle coordinate curvilinee d'una superficie e dello spazio Ⓣ, a five part paper the first of which appeared in 1867, the second and third parts in the following year, with the final two parts in 1870 and 1871. Although, as we mentioned above, Codazzi's submission for the 1859 Grand Prix was not published until ten years after his death, nevertheless his famous formulas appear in this five part paper. He remained at the University of Pavia for the rest of his life, although he only lived for a further eight years after his appointment to the chair of algebra and analytic geometry. He did, however, hold the chair of theoretical geodesics at Pavia for a while.
Codazzi also published on isometric lines, geodesic triangles and the stability of floating bodies.
Article by: J J O'Connor and E F Robertson