International Journal of Mathematics and Mathematical Sciences

Volume 2006 (2006), Article ID 56397, 51 pages

http://dx.doi.org/10.1155/IJMMS/2006/56397

## Connection theory on differentiable fibre bundles: A concise introduction

Laboratory of Mathematical Modeling in Physics, Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Boulivare Tzarigradsko Chaussée 72, Sofia 1784, Bulgaria

Received 5 May 2006; Revised 27 July 2006; Accepted 8 August 2006

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- D. V. Alekseevskii, A. M. Vinogradov, and V. V. Lychagin, “Basic ideas and concepts of differential geometry,” in
*Science and Technology Reviews*, vol. 28 of*Modern Problems in Mathematics, Fundamental Directions. Geometry-1*, pp. 5–298, VINITI, Moscow, 1988. View at Google Scholar - M. F. Atiyah,
*$K$-Theory*, Harvard University, Massachusetts, 1965. - M. Berger,
*A Panoramic View of Riemannian Geometry*, Springer, Berlin, 2003. View at Zentralblatt MATH · View at MathSciNet - R. L. Bishop and R. J. Crittenden,
*Geometry of Manifolds*, vol. 15 of*Pure and Applied Mathematics*, Academic Press, New York, 1964. View at Zentralblatt MATH · View at MathSciNet - Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick,
*Analysis, Manifolds and Physics*, North-Holland, Amsterdam, 1982. View at Zentralblatt MATH · View at MathSciNet - R. Dandoloff and W. J. Zakrzewski, “Parallel transport along a space curve and related phases,”
*Journal of Physics. A. Mathematical and General*, vol. 22, no. 11, pp. L461–L466, 1989. View at Publisher · View at Google Scholar · View at MathSciNet - T. Eguchi, P. B. Gilkey, and A. J. Hanson, “Gravitation, gauge theories and differential geometry,”
*Physics Reports*, vol. 66, no. 6, pp. 213–393, 1980. View at Publisher · View at Google Scholar · View at MathSciNet - M. Göckeler and T. Schücker,
*Differential Geometry, Gauge Theories, and Gravity*, Cambridge University Press, Cambridge, 1987. View at Zentralblatt MATH · View at MathSciNet - W. Greub, S. Halperin, and R. Vanstone,
*Connections, Curvature, and Cohomology. Vol. I: De Rham Cohomology of Manifolds and Vector Bundles*, Academic Press, New York, 1972. View at Zentralblatt MATH · View at MathSciNet - W. Greub, S. Halperin, and R. Vanstone,
*Connections, Curvature, and Cohomology. Vol. II: Lie Groups, Principal Bundles, and Characteristic Classes*, Academic Press, New York, 1973. View at Zentralblatt MATH · View at MathSciNet - W. Greub, S. Halperin, and R. Vanstone,
*Connections, Curvature, and Cohomology. Vol. III: Cohomology of Principal Bundles and Homogeneous Spaces*, Academic Press, New York, 1976. View at Zentralblatt MATH · View at MathSciNet - Ph. Hartman,
*Ordinary Differential Equations*, John Wiley & Sons, New York, 1964. View at Zentralblatt MATH · View at MathSciNet - R. Hermann,
*Vector Bundles in Mathematical Physics. Vol. I*, W. A. Benjamin, New York, 1970. - N. J. Hicks,
*Notes on Differential Geometry*, Van Nostrand Mathematical Studies, no. 3, D. Van Nostrand, New Jersey, 1965. View at Zentralblatt MATH · View at MathSciNet - D. Husemoller,
*Fibre Bundles*, McGraw-Hill, New York, 1966, Russian translation: Mir, Moscow, 1970. View at Zentralblatt MATH · View at MathSciNet - B. Z. Iliev, “Normal frames and the validity of the equivalence principle. I. Cases in a neighbourhood and at a point,”
*Journal of Physics. A. Mathematical and General*, vol. 29, no. 21, pp. 6895–6901, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Z. Iliev, “Normal frames for derivations and linear connections and the equivalence principle,”
*Journal of Geometry and Physics*, vol. 45, no. 1-2, pp. 24–53, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Kobayashi, “Theory of connections,”
*Annali di Matematica Pura ed Applicata. Serie Quarta*, vol. 43, pp. 119–194, 1957. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Kobayashi and K. Nomizu,
*Foundations of Differential Geometry. Vol I*, John Wiley & Sons, New York, 1963, Russian translation: Nauka, Moscow, 1981. View at Zentralblatt MATH · View at MathSciNet - S. Kobayashi and K. Nomizu,
*Foundations of Differential Geometry. Vol. II*, vol. II of*Interscience Tracts in Pure and Applied Mathematics, no. 15*, John Wiley & Sons, New York, 1969. View at Zentralblatt MATH · View at MathSciNet - I. Kolář, P. W. Michor, and J. Slovák,
*Natural Operations in Differential Geometry*, Springer, Berlin, 1993. View at Zentralblatt MATH · View at MathSciNet - A. Lichnerowicz,
*Global Theory of Connections and Holonomy Groups*, Foreign Literature, Moscow, 1960, Russian translation from the French original: Théorie globale des connexions et des groupes d'holonomie, Edizioni Cremonese, Roma, 1955. - Ü. G. Lumiste, “Connection theory in fibre bundles,” in
*Science Review*, Mathematics: Algebra. Topology. Geometry. 1969, pp. 123–168, VINITI, Moscow, 1971. View at Google Scholar - L. Mangiarotti and G. Sardanashvily,
*Connections in Classical and Quantum Field Theory*, World Scientific, New Jersey, 2000. View at Zentralblatt MATH · View at MathSciNet - P. W. Michor,
*Gauge Theory for Fiber Bundles*, vol. 19 of*Monographs and Textbooks in Physical Science. Lecture Notes*, Bibliopolis, Naples, 1991. View at Zentralblatt MATH · View at MathSciNet - A. S. Mishchenko,
*Vector Fibre Bundles and Their Applications*, Nauka, Moscow, 1984. View at Zentralblatt MATH · View at MathSciNet - C. Nash and S. Sen,
*Topology and Geometry for Physicists*, Academic Press, London, 1983. View at Zentralblatt MATH · View at MathSciNet - L. I. Nicolaescu,
*Lectures on the Geometry of Manifolds*, World Scientific, New Jersey, 1st edition, 1996, 4th ed., World Scientific, 2006. View at Zentralblatt MATH · View at MathSciNet - W. A. Poor,
*Differential Geometric Structures*, McGraw-Hill, New York, 1981. View at Zentralblatt MATH · View at MathSciNet - M. Rahula,
*New Problems in Differential Geometry*, vol. 8 of*Series on Soviet and East European Mathematics*, World Scientific, New Jersey, 1993. View at Zentralblatt MATH · View at MathSciNet - R. K. Sachs and H. H. Wu,
*General Relativity for Mathematicians*, Springer, New York, 1977. View at Zentralblatt MATH · View at MathSciNet - D. J. Saunders,
*The Geometry of Jet Bundles*, vol. 142 of*London Mathematical Society Lecture Note Series*, Cambridge University Press, Cambridge, 1989. View at Zentralblatt MATH · View at MathSciNet - B. F. Schutz,
*Geometrical Methods of Mathematical Physics*, Cambridge University Press, Cambridge, 1982, Russian translation: Mir, Moscow, 1984. - M. Spivak,
*A Comprehensive Introduction to Differential Geometry*, vol. 1, Brandeis University, Massachusetts, 1970. View at Zentralblatt MATH · View at MathSciNet - N. Steenrod,
*The Topology of Fibre Bundles*, Princeton University Press, New Jersey, 9th edition, 1974. - S. Sternberg,
*Lectures on Differential Geometry*, Prentice-Hall, New Jersey, 1st edition, 1964. View at Zentralblatt MATH · View at MathSciNet - S. Sternberg,
*Lectures on Differential Geometry*, Chelsea, New York, 2nd edition, 1983, Russian translation: Mir, Moscow, 1970. View at Zentralblatt MATH · View at MathSciNet - R. Sulanke and P. Wintgen,
*Differential Geometry and Fibre Bundles*, Mir, Moscow, 1975. - I. Tamura,
*Topology of Foliations*, Mir, Moscow, 1979, Russian translation from 1976 Japanese original. View at Zentralblatt MATH · View at MathSciNet - F. W. Warner,
*Foundations of Differentiable Manifolds and Lie Groups*, vol. 94 of*Graduate Texts in Mathematics*, Springer, New York, 1983, Russian translation: Mir, Moscow, 1987. View at Zentralblatt MATH · View at MathSciNet - K. Yano,
*The Theory of Lie Derivatives and Its Applications*, North-Holland, Amsterdam, 1957. View at Zentralblatt MATH · View at MathSciNet - K. Yano and M. Kon,
*Structures on Manifolds*, vol. 3 of*Series in Pure Mathematics*, World Scientific, Singapore, 1984. View at Zentralblatt MATH · View at MathSciNet