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June  2022, 14(2): 307-329. doi: 10.3934/jgm.2021027

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 Published  June 2022 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, 2022, 14 (2) : 307-329. 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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[12]

M. Heuer and M. Jotz Lean, Multiple vector bundles: Cores, splittings and decompositions, Theory Appl. Categ., 35 (2020), 665–699. Available from: http://www.tac.mta.ca/tac/volumes/35/19/35-19.pdf.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[24]

J. Pike, Weil Algebras and Double Lie Algebroids, Ph.D. thesis, University of Toronto, 2020. Available from: https://tspace.library.utoronto.ca/bitstream/1807/103377/4/Pike_Jeffrey_202011_PhD_thesis.pdf.

[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.

[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. 

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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[12]

M. Heuer and M. Jotz Lean, Multiple vector bundles: Cores, splittings and decompositions, Theory Appl. Categ., 35 (2020), 665–699. Available from: http://www.tac.mta.ca/tac/volumes/35/19/35-19.pdf.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[24]

J. Pike, Weil Algebras and Double Lie Algebroids, Ph.D. thesis, University of Toronto, 2020. Available from: https://tspace.library.utoronto.ca/bitstream/1807/103377/4/Pike_Jeffrey_202011_PhD_thesis.pdf.

[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.

[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. 

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