American Institute of Mathematical Sciences

Advanced
doi: 10.3934/jgm.2021027
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Quotients of double vector bundles and multigraded bundles

Eckhard Meinrenken ,

Mathematics Department, University of Toronto, 40 St George Street, Toronto, ON M5S 2E4, Canada

* Corresponding author: Eckhard Meinrenken

Received  December 2020 Early access October 2021

Fund Project: The author is supported by NSERC Discovery Grant 480547

We study quotients of multi-graded bundles, including double vector bundles. Among other things, we show that any such quotient fits into a tower of affine bundles. Applications of the theory include a construction of normal bundles for weighted submanifolds, as well as for pairs of submanifolds with clean intersection.

Keywords: Double structures, double vector bundles, graded bundles, clean intersections, weightings.
Mathematics Subject Classification: Primary: 53C05, 53D17; Secondary: 55R65.
Citation: Eckhard Meinrenken. Quotients of double vector bundles and multigraded bundles. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021027
References:
[1]

Z. Chen, Z. J. Liu and Y. H. Sheng, On double vector bundles, Acta Math. Sin. (Engl. Ser.), 30 (2014), 1655-1673.  doi: 10.1007/s10114-014-2412-4.  Google Scholar

[2]

F. del Carpio-Marek, Geometric Structures on Degree $2$ Manifolds, Ph.D. thesis, IMPA, 2015. Google Scholar

[3]

C. Ehresmann, Prolongements des catégories différentiables, in Topologie et Géométrie Différentielle (Séminaire Ehresmann, Vol. VI, 1964), Inst. Henri Poincaré, Paris, 1964, 8pp.  Google Scholar

[4]

V. Fischer and M. Ruzhansky, Quantization on Nilpotent Lie Groups, Progress in Mathematics, 314, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-29558-9.  Google Scholar

[5]

M. K. Flari and K. Mackenzie, Warps, grids and curvature in triple vector bundles, Lett. Math. Phys., 109 (2019), 135-185.  doi: 10.1007/s11005-018-1103-y.  Google Scholar

[6]

J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys., 62 (2012), 21-36.  doi: 10.1016/j.geomphys.2011.09.004.  Google Scholar

[7]

J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305.  doi: 10.1016/j.geomphys.2009.06.009.  Google Scholar

[8]

A. Gracia-Saz and K. C. H. Mackenzie, Duality functors for $n$-fold vector bundles, preprint, arXiv: 1209.0027. Google Scholar

[9]

A. Gracia-Saz and K. C. H. Mackenzie, Duality functors for triple vector bundles, Lett. Math. Phys., 90 (2009), 175-200.  doi: 10.1007/s11005-009-0346-z.  Google Scholar

[10]

A. Gracia-Saz and R. A. Mehta, Lie algebroid structures on double vector bundles and representation theory of Lie algebroids, Adv. Math., 223 (2010), 1236-1275.  doi: 10.1016/j.aim.2009.09.010.  Google Scholar

[11]

A. Gracia-Saz and R. A. Mehta, $\mathcal VB$-groupoids and representation theory of Lie groupoids, J. Symplectic Geom., 15 (2017), 741-783.  doi: 10.4310/JSG.2017.v15.n3.a5.  Google Scholar

[12]

M. Heuer and M. Jotz Lean, Multiple vector bundles: Cores, splittings and decompositions, Theory Appl. Categ., 35 (2020), 665-699.   Google Scholar

[13]

I. Kolář, P. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02950-3.  Google Scholar

[14]

Y. Loizides and E. Meinrenken, Differential geometry of weightings, preprint, arXiv: 2010.01643. Google Scholar

[15]

K. C. H. Mackenzie, Double Lie algebroids and second-order geometry. I., Adv. Math., 94 (1992), 180-239.  doi: 10.1016/0001-8708(92)90036-K.  Google Scholar

[16]

K. C. H. Mackenzie, Duality and triple structures, in The Breadth of Symplectic and Poisson Geometry, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005,455–481. doi: 10.1007/0-8176-4419-9_15.  Google Scholar

[17]

K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar

[18]

K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452.  doi: 10.1215/S0012-7094-94-07318-3.  Google Scholar

[19]

E. Meinrenken, Euler-like vector fields, normal forms, and isotropic embeddings, Indag. Math. (N.S.), 32 (2021), 224-245.  doi: 10.1016/j.indag.2020.08.006.  Google Scholar

[20]

E. Meinrenken and J. Pike, The Weil algebra for double Lie algebroids, Int. Math. Res. Not. IMRN, 2021 (2021), 8550-8622.  doi: 10.1093/imrn/rnz361.  Google Scholar

[21]

R. B. Melrose, Differential Analysis on Manifolds with Corners, manuscript. Available from: http://www-math.mit.edu/ rbm/book.html. Google Scholar

[22]

A. Morimoto, Liftings of tensor fields and connections to tangent bundles of higher order, Nagoya Math. J., 40 (1970), 99-120.  doi: 10.1017/S002776300001388X.  Google Scholar

[23]

T. Nagano, $1$-forms with the exterior derivative of maximal rank, J. Differential Geometry, 2 (1968), 253-264.  doi: 10.4310/jdg/1214428439.  Google Scholar

[24]

J. Pike, Weil Algebras and Double Lie Algebroids, Ph.D. thesis, University of Toronto, 2020.  Google Scholar

[25]

J. Pradines, Fibres Vectoriels Doubles et Calcul des Jets non Holonomes, Esquisses Mathématiques, 29, Université d'Amiens, U.E.R. de Mathématiques, Amiens, 1977.  Google Scholar

[26]

J. Pradines, Représentation des jets non holonomes par des morphismes vectoriels doubles soudés, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 1523-1526.   Google Scholar

show all references

References:
[1]

Z. Chen, Z. J. Liu and Y. H. Sheng, On double vector bundles, Acta Math. Sin. (Engl. Ser.), 30 (2014), 1655-1673.  doi: 10.1007/s10114-014-2412-4.  Google Scholar

[2]

F. del Carpio-Marek, Geometric Structures on Degree $2$ Manifolds, Ph.D. thesis, IMPA, 2015. Google Scholar

[3]

C. Ehresmann, Prolongements des catégories différentiables, in Topologie et Géométrie Différentielle (Séminaire Ehresmann, Vol. VI, 1964), Inst. Henri Poincaré, Paris, 1964, 8pp.  Google Scholar

[4]

V. Fischer and M. Ruzhansky, Quantization on Nilpotent Lie Groups, Progress in Mathematics, 314, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-29558-9.  Google Scholar

[5]

M. K. Flari and K. Mackenzie, Warps, grids and curvature in triple vector bundles, Lett. Math. Phys., 109 (2019), 135-185.  doi: 10.1007/s11005-018-1103-y.  Google Scholar

[6]

J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures, J. Geom. Phys., 62 (2012), 21-36.  doi: 10.1016/j.geomphys.2011.09.004.  Google Scholar

[7]

J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305.  doi: 10.1016/j.geomphys.2009.06.009.  Google Scholar

[8]

A. Gracia-Saz and K. C. H. Mackenzie, Duality functors for $n$-fold vector bundles, preprint, arXiv: 1209.0027. Google Scholar

[9]

A. Gracia-Saz and K. C. H. Mackenzie, Duality functors for triple vector bundles, Lett. Math. Phys., 90 (2009), 175-200.  doi: 10.1007/s11005-009-0346-z.  Google Scholar

[10]

A. Gracia-Saz and R. A. Mehta, Lie algebroid structures on double vector bundles and representation theory of Lie algebroids, Adv. Math., 223 (2010), 1236-1275.  doi: 10.1016/j.aim.2009.09.010.  Google Scholar

[11]

A. Gracia-Saz and R. A. Mehta, $\mathcal VB$-groupoids and representation theory of Lie groupoids, J. Symplectic Geom., 15 (2017), 741-783.  doi: 10.4310/JSG.2017.v15.n3.a5.  Google Scholar

[12]

M. Heuer and M. Jotz Lean, Multiple vector bundles: Cores, splittings and decompositions, Theory Appl. Categ., 35 (2020), 665-699.   Google Scholar

[13]

I. Kolář, P. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02950-3.  Google Scholar

[14]

Y. Loizides and E. Meinrenken, Differential geometry of weightings, preprint, arXiv: 2010.01643. Google Scholar

[15]

K. C. H. Mackenzie, Double Lie algebroids and second-order geometry. I., Adv. Math., 94 (1992), 180-239.  doi: 10.1016/0001-8708(92)90036-K.  Google Scholar

[16]

K. C. H. Mackenzie, Duality and triple structures, in The Breadth of Symplectic and Poisson Geometry, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005,455–481. doi: 10.1007/0-8176-4419-9_15.  Google Scholar

[17]

K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar

[18]

K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452.  doi: 10.1215/S0012-7094-94-07318-3.  Google Scholar

[19]

E. Meinrenken, Euler-like vector fields, normal forms, and isotropic embeddings, Indag. Math. (N.S.), 32 (2021), 224-245.  doi: 10.1016/j.indag.2020.08.006.  Google Scholar

[20]

E. Meinrenken and J. Pike, The Weil algebra for double Lie algebroids, Int. Math. Res. Not. IMRN, 2021 (2021), 8550-8622.  doi: 10.1093/imrn/rnz361.  Google Scholar

[21]

R. B. Melrose, Differential Analysis on Manifolds with Corners, manuscript. Available from: http://www-math.mit.edu/ rbm/book.html. Google Scholar

[22]

A. Morimoto, Liftings of tensor fields and connections to tangent bundles of higher order, Nagoya Math. J., 40 (1970), 99-120.  doi: 10.1017/S002776300001388X.  Google Scholar

[23]

T. Nagano, $1$-forms with the exterior derivative of maximal rank, J. Differential Geometry, 2 (1968), 253-264.  doi: 10.4310/jdg/1214428439.  Google Scholar

[24]

J. Pike, Weil Algebras and Double Lie Algebroids, Ph.D. thesis, University of Toronto, 2020.  Google Scholar

[25]

J. Pradines, Fibres Vectoriels Doubles et Calcul des Jets non Holonomes, Esquisses Mathématiques, 29, Université d'Amiens, U.E.R. de Mathématiques, Amiens, 1977.  Google Scholar

[26]

J. Pradines, Représentation des jets non holonomes par des morphismes vectoriels doubles soudés, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 1523-1526.   Google Scholar

[1]

Guillermo Dávila-Rascón, Yuri Vorobiev. Hamiltonian structures for projectable dynamics on symplectic fiber bundles. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1077-1088. doi: 10.3934/dcds.2013.33.1077
[2]

Jordi-Lluís Figueras, Àlex Haro. Triple collisions of invariant bundles. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2069-2082. doi: 10.3934/dcdsb.2013.18.2069
[3]

Bas Janssens. Infinitesimally natural principal bundles. Journal of Geometric Mechanics, 2016, 8 (2) : 199-220. doi: 10.3934/jgm.2016004
[4]

V. Balaji, I. Biswas and D. S. Nagaraj. Principal bundles with parabolic structure. Electronic Research Announcements, 2001, 7: 37-44.

[5]

J. B. van den Berg, J. D. Mireles James. Parameterization of slow-stable manifolds and their invariant vector bundles: Theory and numerical implementation. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4637-4664. doi: 10.3934/dcds.2016002
[6]

Hideyuki Suzuki, Shunji Ito, Kazuyuki Aihara. Double rotations. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 515-532. doi: 10.3934/dcds.2005.13.515
[7]

Thorsten Hüls. Computing stable hierarchies of fiber bundles. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3341-3367. doi: 10.3934/dcdsb.2017140
[8]

Mauro Patrão, Luiz A. B. San Martin. Morse decomposition of semiflows on fiber bundles. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 561-587. doi: 10.3934/dcds.2007.17.561
[9]

Peter Albers, Jean Gutt, Doris Hein. Periodic Reeb orbits on prequantization bundles. Journal of Modern Dynamics, 2018, 12: 123-150. doi: 10.3934/jmd.2018005
[10]

Antonio Pumariño, Claudia Valls. On the double pendulum: An example of double resonant situations. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 413-448. doi: 10.3934/dcds.2004.11.413
[11]

Marco Castrillón López, Pablo M. Chacón, Pedro L. García. Lagrange-Poincaré reduction in affine principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 399-414. doi: 10.3934/jgm.2013.5.399
[12]

Alexandra Monzner, Nicolas Vichery, Frol Zapolsky. Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization. Journal of Modern Dynamics, 2012, 6 (2) : 205-249. doi: 10.3934/jmd.2012.6.205
[13]

Javier Fernández, Marcela Zuccalli. A geometric approach to discrete connections on principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 433-444. doi: 10.3934/jgm.2013.5.433
[14]

Marco Castrillón López, Pedro Luis García Pérez. The problem of Lagrange on principal bundles under a subgroup of symmetries. Journal of Geometric Mechanics, 2019, 11 (4) : 539-552. doi: 10.3934/jgm.2019026
[15]

Radu Pantilie. On the embeddings of the Riemann sphere with nonnegative normal bundles. Electronic Research Announcements, 2018, 25: 87-95. doi: 10.3934/era.2018.25.009
[16]

Santiago Cañez. Double groupoids and the symplectic category. Journal of Geometric Mechanics, 2018, 10 (2) : 217-250. doi: 10.3934/jgm.2018009
[17]

Michael Hutchings, Frank Morgan, Manuel Ritore and Antonio Ros. Proof of the double bubble conjecture. Electronic Research Announcements, 2000, 6: 45-49.

[18]

Joel Hass, Michael Hutchings and Roger Schlafly. The double bubble conjecture. Electronic Research Announcements, 1995, 1: 98-102.

[19]

Henk Bruin, Gregory Clack. Inducing and unique ergodicity of double rotations. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4133-4147. doi: 10.3934/dcds.2012.32.4133
[20]

Sergiu Aizicovici, Nikolaos S. Papageorgiou, Vasile Staicu. Nonlinear Dirichlet problems with double resonance. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1147-1168. doi: 10.3934/cpaa.2017056

2020 Impact Factor: 0.857

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy