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Topological methods in the theory of Lebesgue area. (English) Zbl 0038.20204


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[1] Henri Cartan, Méthodes modernes en topologie algébrique, Comment. Math. Helv. 18 (1945), 1 – 15 (French). · Zbl 0061.40502 · doi:10.1007/BF02568096
[2] Lamberto Cesari, Su di un problema di analysis situs dello spazio ordinario, Ist. Lombardo Sci. Lett. Cl. Sci. Mat. Nat. Rend. (3) 6(75) (1942), 267 – 291 (Italian). · Zbl 0027.09502
[3] Lamberto Cesari, Caratterizzazione analitica delle superficie continue di area finita secondo Lebesgue, Ann. Scuola Norm. Super. Pisa (2) 11 (1942), 1 – 42 (Italian). · Zbl 0027.20604
[4] Lamberto Cesari, Sui fondamenti geometrici dell’integrale classico per l’area delle superficie in forma parametrica, Atti Accad. Italia. Mem. Cl. Sci. Fis. Mat. Nat. 13 (1943), 1323 – 1481 (Italian). · Zbl 0061.11001
[5] Lamberto Cesari, Sulle superficie di area finita secondo Lebesgue, Atti Accad. Italia. Rend. Cl. Sci. Fis. Mat. Nat. (7) 3 (1942), 350 – 365 (Italian). · Zbl 0027.30202
[6] Lamberto Cesari, Sul concetto di trasformazione assolutamente continua, Boll. Un. Mat. Ital. (2) 5 (1943), 5 – 10 (Italian). · Zbl 0027.38903
[7] Lamberto Cesari, Una uguaglianza fondamentale per l’area delle superficie, Atti Accad. Italia. Mem. Cl. Sci. Fis. Mat. Nat. 14 (1944), 891 – 951 (Italian). · Zbl 0061.11103
[8] L. Cesari, Rappresentazione quasi conforme delle superficie continue, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1 (1946), 509 – 514 (Italian). · Zbl 0061.11102
[9] Lamberto Cesari, Sulla rappresentazione delle superficie continue di area finita secondo Lebesgue, Ist. Lombardo Sci. Lett. Cl. Sci. Mat. Nat. Rend. (3) 10(79) (1946), 31 (Italian). · Zbl 0061.11101
[10] Samuel Eilenberg, On a linkage theorem by L. Cesari, Bull. Amer. Math. Soc. 53 (1947), 1192 – 1195. · Zbl 0032.31504
[11] M. Fréchet, Sur la distance de deux surfaces, Annales de la Société Polonaise Mathématique vol. 3 (1924) pp. 4-19. · JFM 51.0455.02
[12] Z. de Geöcze, Recherches générales sur la quadrature des surfaces courbes, part I, Math. Naturw. Berichte Ungarn vol. 27 (1909) pp. 1-21; part II, ibid. pp. 131-163; part III, ibid. vol. 30 (1912) pp. 1-29. · JFM 45.0451.02
[13] R. G. Helsel, A theorem on surface area, Trans. Amer. Math. Soc. 61 (1947), 443 – 453. · Zbl 0033.11001
[14] B. v. Kerékjártó, Involutions et surfaces continues, Acta Univ. Szeged, vol. 3 (1927) pp. 49-67. · JFM 53.0555.01
[15] H. Lebesgue, Intégrale, longueur, aire, Ann. Mat. Pura Appl. vol. 7 (1902) pp. 231-359. · JFM 33.0307.02
[16] E. J. McShane, On the semi-continuity of double integrals in the calculus of variations, Ann. of Math. (2) 33 (1932), no. 3, 460 – 484. · Zbl 0004.35401 · doi:10.2307/1968529
[17] Charles B. Morrey Jr., An Analytic Characterization of Surfaces of Finite Lebesgue Area. Part I, Amer. J. Math. 57 (1935), no. 3, 692 – 702. · Zbl 0012.20404 · doi:10.2307/2371197
[18] Tibor Radó, Über das Flächenmaß rektifizierbarer Flächen, Math. Ann. 100 (1928), no. 1, 445 – 479 (German). · JFM 54.0262.01 · doi:10.1007/BF01448856
[19] T. Radó, On continuous transformations in the plane, Fund. Math, vol. 27 (1936) pp. 201-211. · JFM 62.0809.01
[20] Tibor Radó, On absolutely continuous transformations in the plane, Duke Math. J. 4 (1938), no. 1, 189 – 221. · Zbl 0019.08801 · doi:10.1215/S0012-7094-38-00415-6
[21] Tibor Radó, On the semi-continuity of double integrals in parametric form, Trans. Amer. Math. Soc. 51 (1942), 336 – 361. · Zbl 0027.40501
[22] Tibor Radó, On continuous path-surfaces of zero area, Ann. of Math. (2) 44 (1943), 173 – 191. · Zbl 0061.10801 · doi:10.2307/1968762
[23] Tibor Radó, On continuous mappings of Peano spaces, Trans. Amer. Math. Soc. 58 (1945), 420 – 454. · Zbl 0061.10802
[24] Tibor Rado, On surface area, Proc. Nat. Acad. Sci. U. S. A. 31 (1945), 102 – 106. · Zbl 0061.10803
[25] Tibor Radó, Two-dimensional concepts of bounded variation and absolute continuity, Duke Math. J. 14 (1947), 587 – 608. · Zbl 0029.35002
[26] Tibor Radó, Length and Area, American Mathematical Society Colloquium Publications, vol. 30, American Mathematical Society, New York, 1948.
[27] T. Radó and P. Reichelderfer, A theory of absolutely continuous transformations in the plane, Trans. Amer. Math. Soc. 49 (1941), 258 – 307. · Zbl 0024.38701
[28] Paul V. Reichelderfer, On bounded variation and absolute continuity for parametric representations of continuous surfaces, Trans. Amer. Math. Soc. 53 (1943), 251 – 291. · Zbl 0061.10602
[29] J. H. Roberts and N. E. Steenrod, Monotone transformations of two-dimensional manifolds, Ann. of Math. (2) 39 (1938), no. 4, 851 – 862. · Zbl 0019.37203 · doi:10.2307/1968468
[30] Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, v. 28, American Mathematical Society, New York, 1942. · Zbl 0061.39301
[31] J. W. T. Youngs, Curves and surfaces, Amer. Math. Monthly 51 (1944), 1 – 11. · Zbl 0061.10902 · doi:10.2307/2303560
[32] J. W. T. Youngs, The topological theory of Fréchet surfaces, Ann. of Math. (2) 45 (1944), 753 – 785. · Zbl 0061.10903 · doi:10.2307/1969303
[33] J. W. T. Youngs, On surfaces of class \?\(_{1}\), Bull. Amer. Math. Soc. 51 (1945), 669 – 673. · Zbl 0061.10904
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