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Karl M. Peterson: The earliest derivation of the Mainardi-Codazzi equations and the fundamental theorem of surface theory. (English) Zbl 0405.01024

Summary: In 1853 Karl M. Peterson, a young Latvian student at the University of Dorpat [Tartu], submitted a candidates’ dissertation [a requirement for obtaining a degree] containing a derivation of the Gauss-Mainardi-Codazzi equations and a proof of the Fundamental Theorem of Surface Theory. This predates the publications of equivalent equations derived by G. Mainardi [1856] and D. Codazzi [1860 and 1868–1869], as well as O. Bonnet’s proof of the Fundamental Theorem. Although Peterson later [1865–1881] played a major role in the formation of a school of differential geometry at Moscow University, this early work remained virtually unknown until its translation [into Russian] and its publication in 1952. The contents of the dissertation and the probable sources of Peterson’s ideas and techniques are discussed.

MSC:

01A60 History of mathematics in the 20th century
53-03 History of differential geometry

Biographic References:

Peterson, Karl M.
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References:

[1] Blank, Ya. P., Klassicheskaya Differentsialnaya Geometriya, (Istoriya Otechestvennoi Matematiki, vol. 3 (1968), Nauka: Nauka Kiev), 371-391
[2] Bonnet, O., Mémoire sur la théorie générale des surfaces, Journal de l’Ecole Polytechnique, 19, 1-146 (1848)
[3] Bonnet, O., Mémoire sur la theorie des surfaces applicables sur une surface donné, Journal de l’Ecole Polytechnique, 25, 31-151 (1867)
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[5] Coolidge, J. L., (A History of Geometrical Methods (1963), Dover) · JFM 66.0685.01
[6] Depman, I. Ya, Karl Mikhailovich Peterson i evo Kandidatskaya dissertatsiya, Istoriko-Matematicheskie Issledovaniya, 5, 134-163 (1952) · Zbl 0049.00502
[7] Egorov, D. V., Raboty K.M. Petersona po teori uravnenii c chastnymi proizvodnymi, Mat. Sb., 24, 22-29 (1903)
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[12] Mainardi, G., Sulla teoria generale delle superficie, Giornale dell’Istituto Lombardo, 9, 385-398 (1856)
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[14] Mlodzeyevskii, B. K., French translation of 1903a, Annales de la Faculté des Sciences de Toulouse, 5, 2, 459-474 (1903)
[15] Mlodzeyevskii, B. K., Ob izgibanii poverkhnostei, Istoriko-Matematicheskie Issledovaniya, 5, 86-112 (1952)
[16] Peterson, K. M., Ob otnosheniyakh i srodstvakh mezhdu krivymi poverkhnostyami, Mat. Sb., 1, 391-438 (1866)
[17] Peterson, K. M., O krivykh na poverkhnostyakh, Mat. Sb., 2, 17-44 (1867)
[18] Peterson, K. M., Ob izgibanni povorkhnostei vtorovo poryadka, Mat. Sb., 10, 476-523 (1882), [Peterson [1866], [1867], and [1882] were translated into French and published in 1905 in Annales de la Faculté des Sciences de Toulouse, together with Peterson’s papers in Differential Equations.]
[19] Peterson, K. M., Ueber Kurven und Flächen (1868), Moscow and Leipzig
[20] Phillips, E. R., Nicolai Nicolaevich Luzin and the Moscow School of the Theory of Functions, Historia Mathematica, 5, 3, 275-305 (1978) · Zbl 0381.01010
[21] Pogorelov, A. V., Differential Geometry (1959), Noordhoff
[22] Rossinskii, S. D., Karl Mikhailovich Peterson (1828-1881), Uspehi Mat. Nauk, 4, 33, 3-13 (1949), 5 · Zbl 0033.00407
[23] Rossinskii, S. D., Kommentarii k dissertatsii K.M. Petersona ‘Ob izgibanii poverkhnostei’, Istoriko-Matematicheskie Issledovaniya, 5, 113-133 (1952)
[24] Stäckel, P., Karl Peterson (1828-1881), Bibliotheca Mathematica, 2, 3, 122-132 (1901) · JFM 32.0010.02
[25] Struik, D., Outline of a History of Differential Geometry, Isis, 20, 161-191 (1934) · JFM 56.1087.05
[26] Struik, D., (Lectures on Classical Differential Geometry (1950), Addison-Wesley) · Zbl 0041.48603
[27] Vucinich, A., (Science in Russian Culture (1970), Stanford University Press), 2 vol.
[28] Yushkevich, A. P., Istoriya Matematiki v Rossi do 1917 goda (1968), Moscow · Zbl 0284.01001
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