Skip to main content
SpringerLink
Log in

Connection theory in bundle spaces

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Literature cited

  1. R. L. Bishop and R. J. Crittenden, Geometry of Manifolds, Academic Press, New York and London (1964).

    Google Scholar 

  2. V. I. Bliznikas, “Affine connection in the space of support elements,” Reports Third Siberian Conf. Math. Mech. [in Russian], Tomsk Univ., Tomsk (1964), pp. 181–183.

    Google Scholar 

  3. V. I. Bliznikas, “On certain manifolds of support elements,” Liet. Mat. Rinkinys, 3(2):221–222 (1963).

    Google Scholar 

  4. V. I. Bliznikas, “The complete object of a centroprojective connection and the torsion-curvature object of a space of central copunctors,” Liet. Mat. Rinkinys, 4(4):457–475 (1964).

    Google Scholar 

  5. V. I. Bliznikas, “On the theory of the curvature of the space of support elements,” Liet. Met. Rinkinys, 5(1):9–24 (1965).

    Google Scholar 

  6. V. I. Bliznikas, “Symmetric spaces of central copunctors,” Liet. Mat. Rinkinys, 5(3):381–389 (1965).

    Google Scholar 

  7. V. I. Bliznikas, “Linear differential-geometric connections of higher-order in the space of support elements,” Izv. Vyssh. Uchebn. Zavedenii, Matematika, No. 5, 13–24 (1966).

    Google Scholar 

  8. V. I. Bliznikas, “Nonholonomic Lie differentiation and linear connections in the space of support elements,” Liet. Mat. Rinkinys, 6(2):141–208 (1966).

    Google Scholar 

  9. V. I. Bliznikas, “On certain connections of bundle spaces,” Liet. Mat. Rinkinys, 7(1):5–16 (1967).

    Google Scholar 

  10. V. V. Vagner, “Absolute derivative of the field of a local geometric object in a composite manifold,” Dokl. Akad. Nauk SSSR, 40(3):94–97 (1943).

    Google Scholar 

  11. V. V. Vagner, “Theory of a composite manifold,” Proc. Seminar Vector and Tensor Analysis, Vol. 8 [in Russian], Moscow State Univ., Moscow (1950), pp. 11–72.

    Google Scholar 

  12. V. V. Vagner, “Algebraic aspects of the general theory of partial connections in bundle spaces,” Izv. Vyssh. Uchebn. Zavedenii, Matematika, No. 11, 26–40 (1968).

    Google Scholar 

  13. V. I. Vedernikov, “Generalization of A. P. Norden's normalization method to the case of a bundle space,” Uch. Zap. Kazansk. Univ., 123(1):3–23 (1963).

    Google Scholar 

  14. V. I. Vedernikov, “Symmetric homogeneous spaces and their role in the theory of Cartan connections,” Reports Third Siberian Conf. Math. Mech. [in Russian], Tomsk Univ., Tomsk (1964), pp. 185–186.

    Google Scholar 

  15. V. I. Vedernikov, “Completely reducible connections,” Izv. Vyssh. Uchebn. Zavedenii, Matematika, No. 1, 33–44 (1965).

    Google Scholar 

  16. V. I. Vedernikov, “Symmetric spaces and conjugate connections,” Uch. Zap. Kazansk. Univ., 125(1):7–59 (1965).

    Google Scholar 

  17. V. I. Vedernikov, “Symmetric spaces. Conjugate connections as a normalized connection,” Tr. Geometr. Seminara, Inst. Nauchn. Inform. Akad. Nauk SSSR, 1:63–88 (1966).

    Google Scholar 

  18. V. A. Gaukhman, “F-structures on principal bundle spaces,” Tr. Tomskogo Univ., 176:20–27 (1964).

    Google Scholar 

  19. L. E. Evtushik, “Nonlinear connections,” Uspekhi Mat. Nauk, 17(2):195–197 (1962).

    Google Scholar 

  20. L. E. Evtushik, “Higher-order nonlinear connections,” Izv. Vyssh. Uchebn. Zavedenii, Matematika, No. 2, 32–44 (1969).

    Google Scholar 

  21. I. L. Kantor, “A generalization of reductive homogeneous spaces,” Dokl. Akad. Nauk SSSR, 151(6):1268–1270 (1963).

    Google Scholar 

  22. I. L. Kantor, “Transitive-differential groups and invariant connections on homogeneous spaces,” Proc. Seminar Vector and Tensor Analysis with Their Appl. to Geom., Mech., and Phys., Issue 13 [in Russian], Moscow Univ., Moscow (1966), pp. 310–398.

    Google Scholar 

  23. É. Cartan, Spaces of Affine, Projective, and Conformal Connection [Russian translation], Izd. Kazansk. Univ., Kazan (1962), 210 pp.

  24. B. L. Laptev, “Covariant differential and the theory of differential invariants in the space of tensor support elements,” Uch. Zap. Kazansk. Univ., 118(4):75–147 (1958).

    Google Scholar 

  25. G. F. Laptev, “Differential connections of manifolds and their holonomy groups,” Dokl. Akad. Nauk SSSR, 71(4):597–600 (1950).

    Google Scholar 

  26. G. F. Laptev, “Manifolds of geometric elements with a differential connection,” Dokl. Akad. Nauk SSSR, 73(1):17–20 (1950).

    Google Scholar 

  27. G. F. Laptev, “Differential geometry of immersed manifolds. Group-theoretical method of differential geometric investigations,” Tr. Mosk. Mat. Obshch., No. 2, 275–382 (1953).

    Google Scholar 

  28. G. F. Laptev, “Group-theoretic method of differential-geometric investigations,” Proc. Third All-Union Math. Congr., 1956, Vol. 3 [in Russian], Akad. Nauk SSSR, Moscow (1958), pp. 409–418.

    Google Scholar 

  29. G. F. Laptev, “Manifolds immersed in generalized spaces,” Proc. Fourth All-Union Math. Congr., 1961, Vol. 2 [in Russian], Nauka, Leningrad (1964), pp. 226–233.

    Google Scholar 

  30. G. F. Laptev, “Fundamental infinitesimal structures of higher orders on a smooth manifold,” Tr. Geometr. Seminara, Inst. Nauchn. Inform. Akad. Nauk SSSR, 1:139–189 (1966).

    Google Scholar 

  31. V. G. Lemlein, “Induction of a connection of constant curvature in associated centroprojective spaces of a locally projective manifold,” Dokl. Akad. Nauk SSSR, 131(1):17–20 (1960).

    Google Scholar 

  32. V. G. Lemlein, “Projective and projective-metric translations in manifolds with affine connection and in Riemann spaces,” Dokl. Akad. Nauk SSSR, 132(6);1261–1264 (1960).

    Google Scholar 

  33. V. G. Lemlein, “Geometric meaning of the projective curvature tensor in manifolds with affine connection,” Dokl. Akad. Nauk SSSR, 133(6):1287–1290 (1960).

    Google Scholar 

  34. V. G. Lemlein, “Spaces with centro-projective connection (Γi jh, Γlm) as manifolds immersed in the representation space of a prolonged pseudogroup of analytic transformations,” Dokl. Akad. Nauk SSSR, 138(6):1291–1294 (1961).

    Google Scholar 

  35. V. G. Lemlein, “Local centro-projective spaces and connections in a differentiable manifold,” Liet. Mat. Rinkinys, 4(1):41–132 (1964).

    Google Scholar 

  36. S. Land, Introduction to Differentiable Manifolds, Interscience Publishers, New York (1962).

    Google Scholar 

  37. A. Lichnerowicz, Théorie Globale des Connexions et des Groupes d'Holonomie, Edizioni Cremonese, Rome (1957).

    Google Scholar 

  38. M. V. Losik “Infinitesimal connections in tangent bundle spaces,” Izv. Vyssh. Uchebn. Zavedenii, Matematika, No. 5, 54–60 (1964).

    Google Scholar 

  39. Ü. G. Lumiste, “Infinitesimal connection in a bundle space of affine frames,” Liet. Mat. Rinkinys, 3(2):211 (1963).

    Google Scholar 

  40. Ü. G. Lumiste, “On the foundations of global connection theory,” Tartu Riikl. Ülikooli Toimetised, No. 150, 69–108 (1964).

    Google Scholar 

  41. Ü. G. Lumiste, “Connection in a bundle space with homogeneous reductive fibers,” Proc. First Republ. Conf. Belorussian Mathematicians, 1964 [in Russian], Minsk (1965), pp. 247–258.

  42. Ü. G. Lumiste, “Connections in homogeneous bundles,” Uspekhi Mat. Nauk, 20(5):263–265 (1965).

    Google Scholar 

  43. Ü. G. Lumiste, “Invariant riggings of a congruence of planes of an affine space,” Izv. Vyssh. Uchebn. Zavedenii, Matematika, No. 6, 93–102 (1965).

    Google Scholar 

  44. Ü. G. Lumiste, “Induced connections in immersed projective and affine bundles,” Tartu Riikl. Ülikooli Toimetised, No. 177, 6–42 (1965(1966)).

    Google Scholar 

  45. Ü. G. Lumiste, “Homogeneous bundles with connection and their immersions,” Tr. Geometr. Seminara, Inst. Nauchn. Inform. Akad. Nauk SSSR, 1:191–237 (1966).

    Google Scholar 

  46. Ü. G. Lumiste, “Connections in homogeneous bundles,” Mat. Sb., 69(3):434–469 (1966).

    Google Scholar 

  47. Ü. G. Lumiste, “On the theory of plane manifolds of a Euclidean space,” Tartu Riikl. Ülikooli Toimetised, No. 192, 12–46 (1966).

    Google Scholar 

  48. Ü. G. Lumiste, “Induced connections in the differential geometry of a family of planes,” Abstracts Reports Third Interinstitutional Conf. Problems Geometry, September 14–19, 1967 [in Russian], Kazan Univ., Kazan (1967), pp. 101–103.

    Google Scholar 

  49. Ü. G. Lumiste, “Stratifiable families of 1-pairs of a four-dimensional projective space,” Tartu Riikl. Ülikooli Toimetised, No. 206, 10–21 (1967).

    Google Scholar 

  50. V. S. Malakhovskii, “Manifolds of geometric elements of genus zero with a fundamental-group connections,” Reports Third Siberian Conf. Math. Mech., 1964 [in Russian], Tomsk Univ., Tomsk (1964), p. 199.

    Google Scholar 

  51. K. Nomizu, Lie Groups and Differential Geometry, The Math. Soc. Japan, Tokyo (1956).

    Google Scholar 

  52. A. L. Onishchik, “Connections without curvature and the De Rham theorem,” Dokl. Akad. Nauk SSSR, 159(5):988–991 (1964).

    Google Scholar 

  53. N. M. Ostianu, “The geometry of a surface of an affine symplectic space,” Uch. Zap. Mosk. Gos. Ped. Inst. im. V. I. Lenina, No. 208, 156–176 (1963).

    Google Scholar 

  54. A. I. Sirota, “The torsion of spaces with infinitesimal connection,” Tbilisis Matematikis Institutis Shromeb. Sakartvelos SSR Metsnierebata Akademia, 27:3–9 (1960).

    Google Scholar 

  55. V. V. Slukhaev, “Nonholonomic manifolds V mn of zero exterior curvature,” Ukrain. Geometr. Sb., No. 4, 78–84 (1967).

    Google Scholar 

  56. C. Teleman, Elemente de Topologie si Varietati Diferentiabile, Editura Didactica si Pedagogica, Bucharest (1964); Russian translation (revised and augmented by the author): Izd. Mir, Moscow (1967).

    Google Scholar 

  57. A. S. Udalov, “On the theory of curves and surfaces in affine and projective spaces,” Izv. Vyssh. Uchebn. Zavedenii, Matematika, No. 4, 158–164 (1963).

    Google Scholar 

  58. A. S. Udalov, “Invariant rigging of surfaces in a projective multidimensional space,” Izv. Vyssh. Uchebn. Zavedenii, Matematika, No. 2, 102–105 (1968).

    Google Scholar 

  59. A. P. Urbonas, “Connections in the space of support elements,” Liet. Mat. Rinkinys, 6(2):279–290 (1966).

    Google Scholar 

  60. S. S. Chern, Complex Manifolds, Textos de Matemática, No. 5, Instituto de Física e Matemática, Universidade do Recife (1959).

  61. Ya. M. Chikvashvili, “Generalization of the Ricci identities for a function of a field of arbitrary type,” Sakartveloc Politekhnikuri Instituti, Shromebi, No. 4(102), 3–9 (1965).

    Google Scholar 

  62. P. I. Shveikin, “Normal geometric objects of a surface in affine space,” Tr. Geometr. Seminara, Inst. Nauchn. Inform. Akad. Nauk SSSR, 1:331–423 (1966).

    Google Scholar 

  63. A. Szybiak, “Lowering of a class of a geometric object,” Zesz. Nauk. Uniw. Jagiell., No. 167, 61–65 (1968).

    Google Scholar 

  64. Yu. Shinkunas, “The space of support linears,” Liet Mat. Rinkinys, 6(3):449–455 (1966).

    Google Scholar 

  65. Yu. Shinkunas, “Connections in spaces of special support elements,” Liet. Mat. Rinkinys, 6(4):622 (1966).

    Google Scholar 

  66. M. Ako, “Non-linear connection in vector bundles,” Kodai Math. Semin. Repts. 18(4):307–316 (1966).

    Google Scholar 

  67. A. Asada, “Connection of topological vector bundles,” Proc. Japan Acad., 41(10):919–922 (1965).

    Google Scholar 

  68. A. Asada, “Connection of topological fibre bundles,” Proc. Japan Acad., 42(1):13–18 (1966).

    Google Scholar 

  69. A. Asada, “Connection of topological fibre bundles. II,” Proc. Japan Acad., 42(3):231–236 (1966).

    Google Scholar 

  70. A. Asada, “Connection of flat vector bundles,” J. Fac. Sci. Shinshu Univ., 2(2):109–116 (1967).

    Google Scholar 

  71. E. Astara, “Sulle connessioni tensoriali decomponibili,” Rend. Semin. Fac. Sci. Univ. Cagliari, 34(1–2):123–134 (1964).

    Google Scholar 

  72. M. F. Atiyah, “Complex analytic connection in fibre bundles,” Trans. Amer. Math. Soc., 85(1):181–207 (1957).

    Google Scholar 

  73. W. Barthel, “Nichtlineare Zusammenhänge und deren Holonomiegruppen,” J. Reine und Angew. Math., 212(3–4):120–149 (1963).

    Google Scholar 

  74. J. P. Benzceri, “Sur la classe d'Euler (ou Stiefel-Whitney) de fibrés affins plats,” C. R. Acad. Sci. 260(21):5442–5444 (1965).

    Google Scholar 

  75. D. Bernard, “Sur la géométrie différentielle des G-structures,” Ann. Inst. Fourier, Vol. 10, 151–270 (1960).

    Google Scholar 

  76. E. Bortolotti, “Connessioni nelle varietà luogo di spazi; applicazione alla geometria metrica differenziale delle congruenze di rette,” Rend. Semin. Fac. Sci. Univ. Cagliari, No. 3, 81–89 (1963).

    Google Scholar 

  77. E. Bortolotti, “Spazi a connessione proiettiva,” Roma, Cremonese (1941).

    Google Scholar 

  78. A. J. Brown, “Connexions on spinor fiber bundles,” Doct. Diss. Univ. Mich., 1966, 138 pp; Dissert. Abstrs., B27(7):2436 (1967).

  79. A. J. Brown, “Geometry of spinors,” J. Math. Anal. and Applic., 25(3):537–555 (1969).

    Google Scholar 

  80. É. Cartan, “Les groupes d'holonomie des espaces généralisés,” Acta math., No. 48, 1–42 (1926).

    Google Scholar 

  81. É. Cartan, “Oeuvre Complètes. II,” Paris (1952),pp. 571–714, 719–856, 1311–1334, 1335–1384.

  82. H. Cartan, “La transgression dans un groupe de Lie et dans un espace fibré principal,” Colloque de Topologie, Bruxelles (1950),pp. 57–71.

  83. I. Cattaneo Gasparini, “Sulle connessioni infinitesimali nello spazio fibrato dei riferimenti affini di una Vn,” Rend. Mat. e Applic., 17(3–4):327–404 (1958).

    Google Scholar 

  84. I. Cattaneo Gasparini, “Sulle connessioni proiettive,” Ann. Mat. Pura ed Appl., Vol. 50, 467–473 (1960).

    Google Scholar 

  85. B. Cenkl, “Les variétés de König généralisées,” Czechoslovak. Math. J., 14(1):1–21 (1964).

    Google Scholar 

  86. B. Cenkl, “L'equation de structure d'un espace a connexion protective,” Czechoslovak. Mat. J., 14(1):79–94 (1964).

    Google Scholar 

  87. B. Cenkl, “Les connexions non-linéaires sur la variete A nm ,” Časop. Pěstov. Mat., 90(1):12–25 (1965).

    Google Scholar 

  88. B. Cenkl, “On the G-structure of higher order,” Časop. Pěstov. Mat., 90(1):26–32 (1965).

    Google Scholar 

  89. B. Cenkl, “Infinitesimal connections of higher order,” Rev. Roumaine Mat. Pures et Appl., 10(9):1277–1280 (1965).

    Google Scholar 

  90. S. S. Chern, Topics in Differential Geometry, Inst. Advanced Study, Princton, New Jersey (1951).

    Google Scholar 

  91. Y. H. Clifton, “On the completeness of Cartan connections,” J. Math. and Mech., 16(6):569–576 (1966).

    Google Scholar 

  92. A. Cossu, “Movimenti speciali in una varietà a connessione tensoriale dotata di transporto Assoluto,” Atti Accad. Naz. Lincei, Rend. Cl. Sci.Fis. Mat, e Natur., 28(2):156–164 (1960).

    Google Scholar 

  93. A. Cossu, “Varietà a connessione tensoriale per tensori doppi controvarianti che ammettono un gruppo d'ordine massimo di movimenti,” Rend. Mat. e Applie., 19(1–2):205–217 (1960).

    Google Scholar 

  94. A. Cossu, “Nozioni generali sulle connessioni tensoriali di specie qualunque,” Rend. Mat. e Applic., 21(1–2):167–218 (1961).

    Google Scholar 

  95. A. Cossu, “Equazioni di struttura di una connessione tensoriale di specie (r, 0),” Rend. Mat. e Applic, 24(3–4):225–241 (1965).

    Google Scholar 

  96. R. Crittenden, “Covariant differentiation,” Quart. J. Math., 13(52):285–298 (1962).

    Google Scholar 

  97. V. Cruceanu, “Sur les espaces à connexion centro-affine,” C. R. Acad. Sci., 250(24):6272–6274 (1965).

    Google Scholar 

  98. V. Cruceanu, “Sur la théorie des variétés plongées dans un espace a connexion métrique projective,” Ann. Stiint. Univ. Iasi, Sec. 1a, 12(1):159–173 (1966).

    Google Scholar 

  99. V. Cruceanu, “Sur certains espaces a connexion centroaffine,” An. Stiint. Univ. Iasi, Sec. 1a, 13(1):69–78 (1967).

    Google Scholar 

  100. V. Cruceanu, “Sur la definition d'une connexion affine,” C. R. Acad. Sci., 266(10):A532-A534 (1968).

    Google Scholar 

  101. C. Di Comite, “Connessioni di ordine n,” Rend. Mat. e Applic., 26(1–2):163–186 (1967).

    Google Scholar 

  102. C. Di Comite, “Sulle connessioni del secondo ordine,” Ann. Mat. Pura ed Appli., Vol. 76, 51–73 (1967).

    Google Scholar 

  103. C. Di Comite, “Pseudoconnessioni tensoriali di specie (r, s) di ordine n,” Ann. Mat. Pura ed Appli., Vol. 79, 107–139 (1968).

    Google Scholar 

  104. C. Ehresmann, “Les connections infinitésimales dans un espace fibré différentiable,” Colloque de Topologie, Bruxelles (1950), pp. 29–55.

  105. C. Ehresmann, “Les prolongements d'une variété différentiable. I. Calcul des jets, prolongement principal. II. L'espace des jets d'ordre r de Vn dans Vm. III. Transitivité des prolongements. IV. Éléments de contact et éléments d'enveloppe. V. Covariants différentiels et prolongements d'une structure infinitésimale,” C. R. Acad. Sci., 233:598–600, 777–779, 1081–1083 (1951); 234: 1028–1030, 1424–1426 (1952).

    Google Scholar 

  106. C. Ehresmann, “Extension du calcul des jets aux jets non holonomes,” C. R. Acad. Sci., 239(25):1762–1764 (1954).

    Google Scholar 

  107. C. Ehresmann, “Applications de la notion de jet non holonome,” C. R. Acad. Sci., 240(4):397–399 (1955).

    Google Scholar 

  108. C. Ehresmann, “Les prolongements d'un espace fibré différentiable,” C. R. Acad. Sci., 240(18):1755–1757 (1955).

    Google Scholar 

  109. C. Ehresmann, “Sur les connections d'ordre supérieur,” Atti del V° Congr. dell' Unione Mat. Ital., 1955, Roma, Cremonese (1956),326 pp.

  110. C. Ehresmann, “Les connexions infinitésimales dans un espace fibré différentiable,” Semin. Bourbaki. Secrét. Math. 1950–51, 2-e année. 2-e éd. Paris, 1959, 24/1–24/16.

  111. H. I. Eliasson, “Geometry of manifolds of maps,” J. Different. Geom., 1(2):169–194 (1967).

    Google Scholar 

  112. F. Fava, “Connessioni tensoriali in spazi proiettivi curvi,” Atti. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Natur, 97:1064–1084 (1962/1963).

    Google Scholar 

  113. F. Fava, “Connessioni tensoriali composte,” Atti Accad. Sci. Torino. Cl. Sci. Fis. Mat. Natur., 98:773–789 (1963/1964).

    Google Scholar 

  114. F. Fava, “Varieta con connessioni tensoriali dotate di autoparallele e loro movimenti,” Rend. Semin. Mat. Univ. e Politechn. Torino, 23:57–73 (1963–1964).

    Google Scholar 

  115. E. A. Feldman, “The geometry of immersions. I,” Trans. Amer. Math. Soc., 120(2):185–224 (1965).

    Google Scholar 

  116. O. Galvani, “Sur la realisation des espaces ponctuels à torsion en géométrie euclidienne,” Ann. Sci. École Norm. Sup.,Vol. 62 (1945), pp. 1–92.

    Google Scholar 

  117. A. Goetz, “A general scheme of inducing infinitesimal connections in principal bundles,” Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. et Phys., 10(1):29–34 (1962).

    Google Scholar 

  118. A. Goetz, “Special connections associated with a given linear connection,” Bull. Acad. Polon. Sci. Sér. Sci. Math., Astron. et Phys., 10(5):277–283 (1962).

    Google Scholar 

  119. A. Goetz, “On induced connections,” Fundam. Math., 55(2):149–174 (1964).

    Google Scholar 

  120. S. Golab, “Sur les comitants algebriques d'un objet de la connexion linéare tensorielle,” Ann. Mat. Pura ed. Appl., Vol. 54, 13–21 (1961).

    Google Scholar 

  121. S. Golab and A. Jakubowicz, “Ein Beitrag zur Theorie des Zusammenhanges für Bivektoren,” Tensor, 20(1):29–34 (1969).

    Google Scholar 

  122. W. Grueb, “Zur Theorie der linearen Übertragungen,” Suomalais. Tiedeakat. Toimituks. (1964), Sar. AI, No. 346, 22 pp.

  123. P. A. Griffiths, “On a theorem of Chern,” Illinois J. Math., 6(3):468–479 (1962).

    Google Scholar 

  124. R. C. Gunning, “Connections for a class of pseudogroup structures,” Proc. Conf. Complex Analysis, Minneapolis, 1964, Berlin-Heidelberg-New York (1965), pp. 186–194.

  125. T. Hangan, “Sur les connexions projectives,” Rev. Mat. Pures et Appl., 3(2):265–276 (1958).

    Google Scholar 

  126. T. Hangan, “Lie derivative and holonomy in the theory of infinitesimal connections,” Rev. Math. Pures et Appl., 5(1):89–102 (1960).

    Google Scholar 

  127. T. Hangan, “Derivata Lie si olonomie in teoria conexinnilor infinitezimale,” Studii si cercetări mat., Acad. RPR, 11(1):159–173 (1960).

    Google Scholar 

  128. T. Hangan, “Transformări projective infinitezimale si olonomie,” Lucrările consf. geometrie si topol., 1958, Acad. RPR (1962),pp. 165–168.

  129. V. Hlavaty, “Affine embedding theory,” Nederl. Akad. Wetensch., Proc. Amsterdam, 52:505–517, 714–724, 977–986 (1949).

    Google Scholar 

  130. Chen-Jung Hsu, “A remark on locally flat infinitesimal connections,” Tohoku Math. J., 11(3):425–429 (1959).

    Google Scholar 

  131. S. Ide, “On the Wirtinger's connections in higher order spaces,” J. Fac. Sci. Hokkaido Univ., Ser 1, Vol. 13, 75–119 (1956).

    Google Scholar 

  132. S. Ide, “On the geometrical meanings of Wirtinger's connections based on Kawaguch's,” Tensor, Vol. 14, 216–218 (1963).

    Google Scholar 

  133. M. Jeger, “Ueber Inflexionen in projektiven Zusammenhaengen,” Univ. e. Politechn. Torino. Rend. Sem. Mat., Vol. 15, 201–224 (1955–1956).

    Google Scholar 

  134. F. Kamber and P. Tondeur, “The characteristic homomorphism of flat bundles,” Topology, 6(2):153–159 (1967).

    Google Scholar 

  135. F. Kamber and P. Tondeur, “Flat bundles and characteristic classes of group-representations,” Amer. J. Math., 89(4):857–886 (1967).

    Google Scholar 

  136. J. Kanitani, “Sur la structure projective d'espace fibré,” Ann. Mat. Pura ed. Appl., Vol. 57, 151–172 (1962).

    Google Scholar 

  137. J. Kanitani, “Sur les arêtes principales relatives à un espace fibré,” J. Math. Kyoto Univ., 4(2):327–353 (1965).

    Google Scholar 

  138. A. Kawaguchi, “On the vectors of higher order and the extended affine connections,” Ann. Mat. Pura ed Appl., Vol. 55, 105–117 (1961).

    Google Scholar 

  139. A. Kawaguchi, “Connections and Their Invariants in Higher Order Spaces,” Perspectives in Geometry and Relativity, Indiana Univ. Press (1966), pp. 192–200.

  140. M. Kavaguchi, “Une observation sur le calcul des calottes,” Tensor, Vol. 14, 182–190 (1963).

    Google Scholar 

  141. S. Kobayashi, “On connections of Cartan,” Canad. J. Math., 8(2):145–156 (1956).

    Google Scholar 

  142. S. Kobayashi, “Theory of connections,” Ann. Mat. Pura ed Appl., Vol. 43, 119–194 (1957).

    Google Scholar 

  143. S. Kobayashi, “On characteristic classes defined by connections,” Tohoku Math. J., 13(3):381–385 (1961).

    Google Scholar 

  144. S. Kobayashi and K. Nomizu, “Foundations of differential geometry. Vol. I,” New York-London, Interscience, (1963), 340 pp.

    Google Scholar 

  145. S. Kobayashi and T. Nagano, “On projective connections,” J. Math. and Mech., 13(2):215–235 (1964).

    Google Scholar 

  146. I. Kolář, “O konexích vyšš⪘h řádu na hlavním fibrovaném prostoru,” Sb. Vojen. Akad. A. Zápoteckého, B17(1):39–47 (1969).

    Google Scholar 

  147. R. König, “Beiträge zu einer allgemeinen Mannigfaltigkeitslehre,” Jahresb. d. Deutsch. Math. Ver., Vol. 28, 213–228 (1920).

    Google Scholar 

  148. J. L. Koszul, “Cohomologie des espaces fibrés différentiables et connexions,” Semin. Bourbaki. Secrét. Math., 1950–1951, 3-e année. 2-e éd. Paris (1959), 38/1–38/7.

  149. J. L. Koszul, “Lectures on fibre bundles and differential geometry,” The Tata Institute of Fundamental Research, Bombay (1960).

    Google Scholar 

  150. M. Kucharzewski, “Über die Tensorübertragung,” Ann. Mat. Pura ed Appl., Vol. 54, 65–83 (1961).

    Google Scholar 

  151. G. Legrand, “Une interpretation de la forme de courbure d'une connexion infinitésimale,” C. R. Acad. Sci., 250(21):3441–3442 (1960).

    Google Scholar 

  152. D. Lehmann, “Plongements de connexions,” C. R. Acad. Sci., 255(14):1566–1568 (1962).

    Google Scholar 

  153. D. Lehmann, “Extensions à courbure nulle d'une connexion,” C. R. Acad. Sci., 258(20):4903–4906 (1964).

    Google Scholar 

  154. D. Lehmann, “Remarques sur la connexion canonique d'une variété de Stifel,” C. R. Acad. Sci., 259(17):2754–2757 (1964).

    Google Scholar 

  155. D. Lehmann, “Une généralisation de la géométrie du plongement,” Cahiers Semin. Topol. et Géom. Différent, Ch. Ehresmann. Fac. Sci, Paris(1964), 6 pp.

  156. D. Lehmann, “Remarque sur les fibrés C∞ de base compacte, admettant une connexion à groupe d'holonomie fini,” C. R. Acad. Sci., 263(10):A348-A351 (1966).

    Google Scholar 

  157. D. Lehmann, “Connexions à courbure nulle sur les fibrés vectoriels de base Pn(R),” C. R. Acad. Sci., 264(10):A443-A445 (1967).

    Google Scholar 

  158. D. Lehmann, “Classification des connexions à courbure nulle,” C. R. Acad. Sci., 266(3):A142-A145 (1968).

    Google Scholar 

  159. D. Lehman, “Quelques propriétés des connexions induites, et fibrés à groupe structural variable en géométrie d'ordre supérieur,” Bull. Soc. Mat. France, Vol.96, Mém. No. 16 (1968), 128 pp.

  160. T. Levi-Civta, “Nozione di parallelismo in una varietà qualunque e consequente specificazione geometrica della curvatura Riemanniana,” Rend. Circ. Matem. Palermo, Vol. 42 (1917), pp. 173–205.

    Google Scholar 

  161. P. Libermann, “On sprays and higher order connections,” Proc. Nat. Acad. Sci. USA, 49(4):459–462 (1963).

    Google Scholar 

  162. P. Libermann, “Surconnexions. Propriétés générales,” C. R. Acad. Sci., 258(26):6327–6330 (1964).

    Google Scholar 

  163. P. Libermann, “Sous-variétés géodésiques et asymptotiques relativement à une surconnexion,” C. R. Acad. Sci., 259(18):2948–2951 (1964).

    Google Scholar 

  164. P. Libermann, “Sur la géométrie des prolongements des espaces fibrés vectoriels,” Ann. Inst. Fourier, 14(1):145–172 (1964).

    Google Scholar 

  165. P. Liberman, “Surconnexions et singularités des applications,” C. R. Acad. Sci., 260(3):776–779 (1965).

    Google Scholar 

  166. P. Libermann, “Surconnexions et connexions affines spéciales,” C. R. Aead. Sci., 261(15):2801–2804 (1965).

    Google Scholar 

  167. P. Libermann, “Calcul tensoriel et connexions d'ordre supérieur,” An. Acad. Brasil Cienc., 37(1):17–29 (1965).

    Google Scholar 

  168. P. Libermann, “Sur le tenseur de structure d'ordre supérieur,” C. R. Acad. Sci., 265(22):A740-A743 (1967).

    Google Scholar 

  169. A. Lichnerowicz, “Groupes d'holonomie,” Proc. Internat. Congr. Mathematicians, 1954, Amsterdam, Vol.1, Groningen-Amsterdam (1957), pp. 334–347.

  170. A. Lichnerowicz, “Transformations des variétés à connexion linéaire et des variétés riemanniennes,” Einseigen. Math. 8(1–2):1–15 (1962).

    Google Scholar 

  171. P. Mastrogiacomo, “Enti geometrici associati ad una connessione tensoriale per tensori controvarianti e covarianti m-pli,” Rend. Mat. e Applic., 18(3–4):426–449 (1959).

    Google Scholar 

  172. P. Mastrogiacomo, “Sul tensore di torsione di una connessione tensoriale per tensori controvarianti e covarianti m-pli,” Riv. Mat. Univ. Parma, 1(2):279–292 (1960).

    Google Scholar 

  173. P. Mastrogiacomo, “Su alcune classi particilari di connessioni tensoriali per tensori controvarianti e covarianti m-pli,” Ann. di Mat. Pura ed Appl.,Vol. 55, 245–272 (1961).

    Google Scholar 

  174. A. Mihai, “Conexiuni meromorfe,” Studii si cercetari mat. Acad. RSR, 20(6):859–865 (1968).

    Google Scholar 

  175. J. Milnor, “On the existence of a connection with curvature zero,” Comment. Math. Helv., 32(3):215–223 (1958).

    Google Scholar 

  176. P. Molino, “Espaces homogénes semi-réductifs et connexione subordonées,” C. R. Acad. Sci., 252(22):3379–3380 (1961).

    Google Scholar 

  177. P. Molino, “Champs d'éléments sur un espace fibré principal différentiable,” Thès. Doct. Sci. Math., Fac. Sci. Univ. Paris, 1963, Luisant-Chartres (1964), 59 pp.

  178. P. Molino, “Champs d'éléments sur un espace fibré principal differentiable,” Ann. Inst. Fourier, 14(2):163–219 (1964).

    Google Scholar 

  179. F. Molino, “Sous-modules transitifs,” Bull. Soc. Math. France, 94(1):15–24 (1966).

    Google Scholar 

  180. A. Morimoto, “Sur la classification des espaces fibrés vectoriels holomorphes sur un tore complexe admettant des connexions holomorphes,” Nagoya Math. J., Vol. 15, 83–154 (1959).

    Google Scholar 

  181. L. Muracchini, “Transformazioni puntuali fra spazi conformi e connessioni conformi,” Boll. Unione Mat. Ital., 17(2):191–198 (1962).

    Google Scholar 

  182. Ch. Murărescu, “La théorie des variétés d'un espace à connexion projective sans torsion. I,” An. Stiint. Univ. Iasi, Sec. la, 12(2):333–339 (1966).

    Google Scholar 

  183. Ch. Murărescu, “Sur la théorie locale des distributions définies sur un espace à connexion projective,” Rev.Roumaine Math. Pures et Appl., 13(7):1001–1007 (1968).

    Google Scholar 

  184. M. S. Narasimhan, “Existence of universal connections,” General Topology and Its Relation to Modern Analysis and Algebra, Academic Press, Prague, Publ. House Czechosl. Acad. Sci. (1962), p. 286

    Google Scholar 

  185. M. S. Narasimhan and S. Ramanan, “Existence of universal connections,” Amer. J. Math., 83(3):563–572 (1961).

    Google Scholar 

  186. M. S. Narasinhan, “Existence of universal connections. II,” Amer. J. Math, 85(2):223–231 (1963).

    Google Scholar 

  187. Ngo Van Que, “De la connexion d'ordre supérieur,” C. R. Acad. Sci., 259(13):2061–2064 (1964).

    Google Scholar 

  188. H. K. Nickerson, “On differential operators and connections,” Trans. Amer. Math. Soc., 99(3):509–539 (1961).

    Google Scholar 

  189. K. Nomizu, “Recent development in the theory of connections and holonomy groups,” Advances Math., 1(1):1–49 (1961).

    Google Scholar 

  190. K. Nomizu, Recent Developments in the Theory of Connections and Holonomy Groups, Academic Press (1962).

  191. K. Nomizu and H. Ozeki, “On the degree of differentiability of curves used in the definition of the holonomy group,” Bull Amer. Math. Soc., 68(2):74–75 (1962).

    Google Scholar 

  192. K. Ogiue, “Theory of conformal connections,” Kodai Math. Semin. Repts, 19(2):193–224 (1967).

    Google Scholar 

  193. V. Oproiu, “Asupra conexiunilor din spatiul fibrat tangent la un spatiu fibrat principal,” Studii si cercetărimat., Acad. RSR, 19(4):531–535 (1967).

    Google Scholar 

  194. V. Oproiu, “On the differential geometry of the tangent bundle,” Rev. Roumaine Math. Pures et Appl. 13(6):847–855 (1968).

    Google Scholar 

  195. Mau Quan Pham, “Introduction a la géométrie des variétés différentiables,” Paris, Dunod, No. XVI, (1969), 278 pp.

    Google Scholar 

  196. E. Picasso, “Varietà a connessione metrica tensoriale,” Atti Accad. Naz. Lincei. Rend. Cl Sci. Fis., Mat. e Natur., 36(6):808–813 (1964).

    Google Scholar 

  197. E. Picasso, “Metriche riemanniane tensoriali,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis., Mat. e Natur., 39(3–4):175–182 (1965).

    Google Scholar 

  198. W. F. Pohl, “Connexions in differential geometry of higher order,” Trans. Amer. Math. Soc., 125(2):310–325 (1966).

    Google Scholar 

  199. T. Postelnicu, “Asupra corespondentelor între două plane proiective,” Lucrările consf. geométrie si topol., 1958, Bucuresti, Acad. RPS (1962), pp. 179–183.

  200. C. Rea, “Sulla riducibilità di una famiglia di connessioni,” An. Scuola Norm. Super. Pisa. Sci. Fis. e Mat., 22(1):31–39 (1968).

    Google Scholar 

  201. T. Saeki, “The holonomy covering space in principal fibre boundles,” Tohoku Math. J., 10(3):304–312 (1958).

    Google Scholar 

  202. S. Sasaki and K. Yano, “On the structure of spaces with normal projective connexions whose groups of holonomy fix a hyperquadric or a quadric of (N-2)-dimension,” Tohoku Math. J., Vol. 1, 31–39 (1949).

    Google Scholar 

  203. H. Sasayama, “On intrinsic derivation of generalized order in the space of R. König's connection,” J. Spat. Math. Sasayama Res. Room, 4(1):1–20 (1961).

    Google Scholar 

  204. H. Sasayama, “On spaces of quasi noneuclidean connections,” J. Spat. Math. Sasayama Res. Room, 8(1–2–3):1–20 (1965).

    Google Scholar 

  205. S. Sato, “On a projective connection and Riemannian metric,” Tensor, Vol. 14, 1–5 (1963).

    Google Scholar 

  206. J. A. Schouten, “Sur les connexions conformes et projectives de M. Cartan et la connexion linéaire générale de M. König,” C. R. Acad. Sci., Vol. 178, 2044–2046 (1924).

    Google Scholar 

  207. J. A. Schouten, “Erlanger Program und Übertragunslehre. Neue Gesichtspunkte zur Grundlegung der Geometrie,” Rend. Circ. Matem. Palermo, Vol. 50, 142–189 (1926).

    Google Scholar 

  208. J. A. Schouten, Ricci-Calculus. An Introduction to Tensor Analysis and It's Geometrical Applications, 2ded., Berlin-Göttingen-Heidelberg, Springer (1954).

    Google Scholar 

  209. A. Seiken, “Connexions in principal fibre bundles with group GL (nm, R),” Doct. Diss., Univ. Michigan, 1963, 64 pp.; Dissert. Abstr., 24(6): 2504 (1963).

  210. I. M. Singer, “The geometric interpretation of a special connection,” Pacif. J. Math., 9(2):585–590 (1959).

    Google Scholar 

  211. W. Slebodzinski, “Sur les prolongements d'une connexion linéaire,” Bull. Acad. Polon. Sci., Sér. Sci. Math., Astron.et Phys., 8(3):145–150 (1960).

    Google Scholar 

  212. F. Speranza, “Determinazione di connessioni prospective,” Boll. Unione Mat. Ital., 18(2):101–107 (1963).

    Google Scholar 

  213. F. Speranza, “Sulle connessioni prospettive. Atti Accad. Sci. Ist. Bologna, C1. Sci. Fis. Rend., 1962–1963, 10(2):91–122 (1963).

    Google Scholar 

  214. F. Speranza, “Sulla nozione di connessione,” Boll. Unione Mat. Ital., 20(3):367–378 (1965).

    Google Scholar 

  215. F. Speranza, “Sui gruppi d'olonomia degli spazi a connessione proiettiva o affine generalizzati,” Boll. Unione Mat. Ital., 21(1):48–64 (1966).

    Google Scholar 

  216. F. Speranza, “Connessioni prospecttive in un sistema di spazi linear,” Period. mat., 46(1–2):326–339 (1968).

    Google Scholar 

  217. M. Stoka, “Corrispondenze tra spazi proiettivi a connessione constante,” Boll. Unione Mat. Ital., 17(1):40–47 (1962).

    Google Scholar 

  218. M. Stoka, “Sur les groupes de mouvement d'une correspondance entre les plans projectifs,” Math. Notae, 18(1):159–184 (1962).

    Google Scholar 

  219. M. Stoka, “Sur les correspondances entre des espaces projectifs à connexion projective linéaire,” Czechoslovak. Math. J., 13(4):622–631 (1963).

    Google Scholar 

  220. M. Stoka, “Asupra corespondentei de speta a treia între plane projective,” Studii si cercetări mat., Acad. RPR, 17(4):547–562 (1965).

    Google Scholar 

  221. A. Švec, “L'application des variétés à connexion à certains problèmes de la géométrie différentielle,” Czechoslovak. Math. J., 10(4):523–550 (1960).

    Google Scholar 

  222. A. Švec, “Les espaces de König du point de vue des espaces fibres à connexion,” Comment. Math. Univ. Carolinae, 3(1):11–14 (1962).

    Google Scholar 

  223. A. Švec, “Le calcul tensoriel généralisé,” Comment. Math. Univ. Carolinae, 1(2):17–22 (1960).

    Google Scholar 

  224. A. Švec, “Au sujet de la definition des variétés de Kön1ig,” Czechoslovak. Math. J., 14(2):222–234 (1964).

    Google Scholar 

  225. A. Švec, “Cartan's method of specialization of frames,” Czechoslovak. Math. J., 16(4):552–599 (1966).

    Google Scholar 

  226. A. Szybiak, “On prolongation of fibre structures and connections of higher order,” Bull. Acad. Polon. Sci., Sér. Sci. Math., Astron. Phys., 13(9):661–664 (1965).

    Google Scholar 

  227. A. Szybiak, “On the general connections and continuity of the operator V,” Bull. Acad. Polon. Sci. Ser. Sci. Math., Astron, et Phys., 13(9):665–667 (1965).

    Google Scholar 

  228. A. Szybiak, “Covariant differentation of geometric objects,” Zesz,Nauk Uniw. Jageill., No. 131, 89–100 (1966).

    Google Scholar 

  229. A. Szybiak, “Covariant differentation of geometric objects,” Rozpr. Mat., No. 56 (1967), 41 pp.

  230. T. Takasu, “Extended affine principal fibre bundles,” Ann. Mat. Pura ed Appl., 54, 85–97 (1961). (1961).

    Google Scholar 

  231. S. Takizawa, “On the induced connections,” Mem. Coll. Sci. Univ. Kyoto, A. Math., 30(2):105–118 (1957).

    Google Scholar 

  232. S. Takizawa, “On soudures of differentiable fibre bundles,” J. Math. Kyoto Univ., 2(3):237–276 (1963).

    Google Scholar 

  233. L. Tamassy, “Über den Affinzusammenhang von zu Tangentialräumen gehörendes Produkträumen,” Acta Math. Acad. Scient. Hung., 11(1–2):65–82 (1960).

    Google Scholar 

  234. L. Tamassy, “Über tensorielle Übertragungen Spezieller,” Art.Publs. Math., 11(1–4):273–277 (1964).

    Google Scholar 

  235. L. Tamassy, “Über autoparallele Flächen tensorial zusammenhängender Räume,” Acta math., Acad. Scient. Hung., 16(1–2):75–87 (1965).

    Google Scholar 

  236. L. Tamassy, “Aus dekomponierbaren Elementen bestehende Gebilde eines Produktraumes,” Mat. Vesn., 2(2):121–126 (1965).

    Google Scholar 

  237. K. Tandai, “On connections of geometric structures,” Proc. Japan. Acad., 44(1):32–34 (1968).

    Google Scholar 

  238. C. Teleman, “Généralisation du groupe fondamental,” Ann. Stiint. École Norm. Super 77(3):195–234 (1960).

    Google Scholar 

  239. C. Teleman, “Sur la classification des espaces fibrés,” C. R. Acad. Sci., 253(6):935–936 (1961).

    Google Scholar 

  240. C. Teleman, “Sur les connexions infinitésimales qu'on peut definir dans les structures fibrées différentiables de base donnée,” Ann. Mat. Pura ed Appl. Vol. 62, 379–412 (1963).

    Google Scholar 

  241. C. Teleman, “Sur une théorie générale des connexions,” Bull. Math. Soc. Sci. Mat. RSR, 10(1–2):179–199 (1966(1967)).

    Google Scholar 

  242. Tran van Tan, “Theorie des connexions spinorielles déduites de connexions euclidiennes,” C. R. Acad. Sci., 253(21):2320–2322 (1961).

    Google Scholar 

  243. I. Vaisman, “A supra geometriei directiilor de pe o varietate diferentiabila,” An. Univ. Ser. Stiinte Mat. Fiz.,Vol. 2, 249–263 (1964).

    Google Scholar 

  244. I. Vaisman, “Courbes, configurations de Myller et distributions dans les espaces à connexion symplectique,” An. Stiint.Univ. Iasi, Sec. Ia, 10(2):417–436 (1964).

    Google Scholar 

  245. I. Vaisman, Contributions à la géométrie différentielle projective-symplectique,” An. Stiint.Univ. Iasi, Sec. 1a, Monogr. No. 1 (1966), 126 pp.

  246. I. Vaisman, “Surfaces holonomes et non holonomes dans les espaces à connexion symplectique à trois dimensions,” An. Stiint. Univ. Iasi, Sec. 1a, 12(1):175–183 (1966).

    Google Scholar 

  247. I. Vaisman, “The curvature groups of a space form,” Ann. Scuola Norm. Super Pisa, Sci. Fize Mat., 22(2):331–341 (1968).

    Google Scholar 

  248. A. Verona, “Connexions généralisées,” Rev. Roumaine Math. Pures et Appl., 13(6):891–896 (1968).

    Google Scholar 

  249. J. Vilms, “Connections on tangent bundles,” J. Different. Geom., 1(3):235–243 (1967).

    Google Scholar 

  250. J. Virsik, “Non-holonomic connections on vector bundles I,” Czechoslovak. Math. J., 17(1):108–147 (1967).

    Google Scholar 

  251. J. Virsik, “Non-holonomic connections on vector bundles, II,” Czechoslovak. Math. J., 17(2):200–224 (1967).

    Google Scholar 

  252. V. Virsik, “Über Zusammenhänge höherer Ordnungen in Vektorraumbüdeln,” Math. Nachr., 34(3–4):219–227 (1967).

    Google Scholar 

  253. J. Virsik, “A generalized point of view to higher order connections on fibre bundles,” Czechoslovak. Math. J., 19(1):110–142 (1969).

    Google Scholar 

  254. V. V. Vagner (Wagner), “Differential geometry of the family of Rk's in Rn and of the family of totally geodesic Sk−1's in Sn−1 of positive curvature,” Mat. Sb., 10(52):165–212 (1942).

    Google Scholar 

  255. V. V. Vagner (Wagner), “Geometria del calcolo delle variazioni,” Centro Internationale Matematico Estivo, 1961, v. II, Roma, Edizioni Cremonese (1965).

    Google Scholar 

  256. H. C. Wang, “Invariant connections over principal fiber bundles,” Nagoya Math. J., Vol. 13, 1–19 (1958).

    Google Scholar 

  257. H. Weyl, Raum, Zeit, Materie, Berlin (1918).

  258. A. P. Whitman and L. Conlon, “A note on holonomy,” Proc. Amer. Math. Soc., 16(5):1046–1051 (1965).

    Google Scholar 

  259. J. A., Wolf, “Differentiable fibre spaces and mappings compatible with Riemannian metrics,” Michigan Math. J., 11(1):65–70 (1964).

    Google Scholar 

Download references

Authors
  1. Ü. G. Lumiste
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Itogi Nauki, Seriya Matematika (Algebra, Topologiya, Geometriya), pp. 123–168, 1969.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lumiste, Ü.G. Connection theory in bundle spaces. J Math Sci 1, 363–390 (1973). https://doi.org/10.1007/BF01083670

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01083670

Keywords

Search

Navigation