June  2016, 8(2): 199-220. doi: 10.3934/jgm.2016004

Infinitesimally natural principal bundles

1. 

Universiteit Utrecht, Budapestlaan 6, 3584 CD Utrecht, Netherlands

Received  June 2015 Revised  April 2016 Published  June 2016

We extend the notion of a natural fibre bundle by requiring diffeomorphisms of the base to lift to automorphisms of the bundle only infinitesimally, i.e. at the level of the Lie algebra of vector fields. We classify the principal fibre bundles with this property. A version of the main result in this paper (theorem 4.4) can be found in Lecomte's work [12]. Our approach was developed independently, uses the language of Lie algebroids, and can be generalized in several directions.
Citation: Bas Janssens. Infinitesimally natural principal bundles. Journal of Geometric Mechanics, 2016, 8 (2) : 199-220. doi: 10.3934/jgm.2016004
References:
[1]

Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-1-4757-6800-8.  Google Scholar

[2]

Math. Zeitschr., 79 (1962), 237-238. doi: 10.1007/BF01193118.  Google Scholar

[3]

Rev. Math. Phys., 13 (2001), 953-1034. doi: 10.1142/S0129055X01000922.  Google Scholar

[4]

Ann. of Math., 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.  Google Scholar

[5]

Proc. London Math. Soc., 38 (1979), 219-236. doi: 10.1112/plms/s3-38.2.219.  Google Scholar

[6]

Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986.  Google Scholar

[7]

H. Glöckner, Differentiable mappings between spaces of sections, 2002., arXiv:1308.1172., ().   Google Scholar

[8]

Transform. Groups, 16 (2011), 137-160. doi: 10.1007/s00031-011-9126-9.  Google Scholar

[9]

Graduate Texts in Mathematics, Springer-Verlag, New York, 1972.  Google Scholar

[10]

Birkhäuser, Boston, first edition, 1996. doi: 10.1007/978-1-4757-2453-0.  Google Scholar

[11]

Princeton University Press, second edition, 1994. Google Scholar

[12]

Bulletin de la S. M. F., 113 (1985), 259-271.  Google Scholar

[13]

Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511661839.  Google Scholar

[14]

Amer. J. Math., 124 (2002), 567-593. doi: 10.1353/ajm.2002.0019.  Google Scholar

[15]

Master's thesis, University of New South Wales, 2001. Google Scholar

[16]

Topology, 39 (2000), 445-467. doi: 10.1016/S0040-9383(98)00069-X.  Google Scholar

[17]

A. Nijenhuis, Theory of the Geometric Object, 1952., Doctoral thesis, ().   Google Scholar

[18]

In Proc. Internat. Congress Math. 1958, pages 463-469. Cambridge Univ. Press, New York, 1960.  Google Scholar

[19]

In Differential geometry (in honor of Kentaro Yano), pages 317-334. Kinokuniya, Tokyo, 1972.  Google Scholar

[20]

Topology, 19 (1977), 271-277.  Google Scholar

[21]

Math. Scand., 8 (1960), 116-120.  Google Scholar

[22]

Comptes Rendus Acad. Sci. Paris A, 263 (1966), 907-910.  Google Scholar

[23]

J. Diff. Geom., 7 (1972), 257-278.  Google Scholar

[24]

Proc. London Math. Soc., S2-42 (1937), 356-376. doi: 10.1112/plms/s2-42.1.356.  Google Scholar

[25]

Proc. Amer. Math. Soc., 5 (1954), 468-472. doi: 10.1090/S0002-9939-1954-0064764-3.  Google Scholar

[26]

Kōji Shiga and Toru Tsujishita, Differential representations of vector fields,, Kōdai Math. Sem. Rep., 28 (): 214.   Google Scholar

[27]

Comp. Math., 26 (1973), 151-158.  Google Scholar

[28]

Am. J. Math., 100 (1978), 775-828. doi: 10.2307/2373910.  Google Scholar

[29]

Abh. Sem. Vektor Tenzoranal. Moskau, 4 (1937), 366-375. Google Scholar

show all references

References:
[1]

Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-1-4757-6800-8.  Google Scholar

[2]

Math. Zeitschr., 79 (1962), 237-238. doi: 10.1007/BF01193118.  Google Scholar

[3]

Rev. Math. Phys., 13 (2001), 953-1034. doi: 10.1142/S0129055X01000922.  Google Scholar

[4]

Ann. of Math., 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.  Google Scholar

[5]

Proc. London Math. Soc., 38 (1979), 219-236. doi: 10.1112/plms/s3-38.2.219.  Google Scholar

[6]

Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986.  Google Scholar

[7]

H. Glöckner, Differentiable mappings between spaces of sections, 2002., arXiv:1308.1172., ().   Google Scholar

[8]

Transform. Groups, 16 (2011), 137-160. doi: 10.1007/s00031-011-9126-9.  Google Scholar

[9]

Graduate Texts in Mathematics, Springer-Verlag, New York, 1972.  Google Scholar

[10]

Birkhäuser, Boston, first edition, 1996. doi: 10.1007/978-1-4757-2453-0.  Google Scholar

[11]

Princeton University Press, second edition, 1994. Google Scholar

[12]

Bulletin de la S. M. F., 113 (1985), 259-271.  Google Scholar

[13]

Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511661839.  Google Scholar

[14]

Amer. J. Math., 124 (2002), 567-593. doi: 10.1353/ajm.2002.0019.  Google Scholar

[15]

Master's thesis, University of New South Wales, 2001. Google Scholar

[16]

Topology, 39 (2000), 445-467. doi: 10.1016/S0040-9383(98)00069-X.  Google Scholar

[17]

A. Nijenhuis, Theory of the Geometric Object, 1952., Doctoral thesis, ().   Google Scholar

[18]

In Proc. Internat. Congress Math. 1958, pages 463-469. Cambridge Univ. Press, New York, 1960.  Google Scholar

[19]

In Differential geometry (in honor of Kentaro Yano), pages 317-334. Kinokuniya, Tokyo, 1972.  Google Scholar

[20]

Topology, 19 (1977), 271-277.  Google Scholar

[21]

Math. Scand., 8 (1960), 116-120.  Google Scholar

[22]

Comptes Rendus Acad. Sci. Paris A, 263 (1966), 907-910.  Google Scholar

[23]

J. Diff. Geom., 7 (1972), 257-278.  Google Scholar

[24]

Proc. London Math. Soc., S2-42 (1937), 356-376. doi: 10.1112/plms/s2-42.1.356.  Google Scholar

[25]

Proc. Amer. Math. Soc., 5 (1954), 468-472. doi: 10.1090/S0002-9939-1954-0064764-3.  Google Scholar

[26]

Kōji Shiga and Toru Tsujishita, Differential representations of vector fields,, Kōdai Math. Sem. Rep., 28 (): 214.   Google Scholar

[27]

Comp. Math., 26 (1973), 151-158.  Google Scholar

[28]

Am. J. Math., 100 (1978), 775-828. doi: 10.2307/2373910.  Google Scholar

[29]

Abh. Sem. Vektor Tenzoranal. Moskau, 4 (1937), 366-375. Google Scholar

[1]

Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021063

[2]

Yusi Fan, Chenrui Yao, Liangyun Chen. Structure of sympathetic Lie superalgebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021020

[3]

F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605

[4]

Fatemeh Abtahi, Zeinab Kamali, Maryam Toutounchi. The BSE concepts for vector-valued Lipschitz algebras. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1171-1186. doi: 10.3934/cpaa.2021011

[5]

Wei Xi Li, Chao Jiang Xu. Subellipticity of some complex vector fields related to the Witten Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021047

[6]

Stephen Doty and Anthony Giaquinto. Generators and relations for Schur algebras. Electronic Research Announcements, 2001, 7: 54-62.

[7]

Julian Koellermeier, Giovanni Samaey. Projective integration schemes for hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (2) : 353-387. doi: 10.3934/krm.2021008

[8]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153

[9]

Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1091-1102. doi: 10.3934/cpaa.2021008

[10]

Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313

[11]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[12]

Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637

[13]

Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79

[14]

Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141

[15]

Zhimin Chen, Kaihui Liu, Xiuxiang Liu. Evaluating vaccination effectiveness of group-specific fractional-dose strategies. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021062

[16]

Fang Li, Jie Pan. On inner Poisson structures of a quantum cluster algebra without coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021021

[17]

Philippe Jouan, Ronald Manríquez. Solvable approximations of 3-dimensional almost-Riemannian structures. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021023

[18]

Lei Lei, Wenli Ren, Cuiling Fan. The differential spectrum of a class of power functions over finite fields. Advances in Mathematics of Communications, 2021, 15 (3) : 525-537. doi: 10.3934/amc.2020080

[19]

A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909

[20]

Ahmad Mousavi, Zheming Gao, Lanshan Han, Alvin Lim. Quadratic surface support vector machine with L1 norm regularization. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021046

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (100)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]