| Title:
|
On curves and jets of curves on supermanifolds (English) |
| Author:
|
Bruce, Andrew James |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
50 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
115-130 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we examine a natural concept of a curve on a supermanifold and the subsequent notion of the jet of a curve. We then tackle the question of geometrically defining the higher order tangent bundles of a supermanifold. Finally we make a quick comparison with the notion of a curve presented here are other common notions found in the literature. (English) |
| Keyword:
|
supermanifolds |
| Keyword:
|
curves |
| Keyword:
|
jets |
| Keyword:
|
higher order tangent bundles |
| MSC:
|
58A20 |
| MSC:
|
58A32 |
| MSC:
|
58A50 |
| idZBL:
|
Zbl 06391571 |
| idMR:
|
MR3215285 |
| DOI:
|
10.5817/AM2014-2-115 |
| . |
| Date available:
|
2014-05-23T09:37:55Z |
| Last updated:
|
2015-03-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143785 |
| . |
| Reference:
|
[1] Alldridge, A.: A convenient category of supermanifolds.arXiv:1109.3161 [math.DG] 2011. |
| Reference:
|
[2] Cariñena, J.F., Figueroa, H.: Geometric formulation of higher order Lagrangian systems in supermechanics.Acta Appl. Math. 51 (1998), 25–58. Zbl 0999.70018, MR 1609857, 10.1023/A:1005870025318 |
| Reference:
|
[3] Carmeli, C., Caston, L., Fioresi, R.: Mathematical Foundations of Supersymmetry.EMS Series of Lectures in Mathematics, European Mathematical Society, 2011. Zbl 1226.58003, MR 2840967 |
| Reference:
|
[4] Deligne, P., Morgan, J.W.: Notes on supersymmetry (following Joseph Bernstein).Quantum fields and strings: a course for mathematicians, Amer. Math. Soc., Providence, RI, 1999, Vol. 1, 2 (Princeton, NJ, 1996/1997), pp. 41–97. Zbl 1170.58302, MR 1701597 |
| Reference:
|
[5] DeWitt, B.: Supermanifolds.Cambridge Monogr. Math. Phys., Cambridge University Press, 1984. Zbl 0551.53002, MR 0778559 |
| Reference:
|
[6] di Bruno, F. Faà: Sullo sviluppo delle Funzioni.Annali di Scienze Matematiche e Fisiche 6 (1855), 479–480, (in Italian). |
| Reference:
|
[7] Dumitrescu, F.: Superconnections and parallel transport.Pacific J. Math. 236 (2008), 307–332. Zbl 1155.58001, MR 2407109, 10.2140/pjm.2008.236.307 |
| Reference:
|
[8] Garnier, S., Wurzbacher, T.: The geodesic flow on a Riemannian supermanifold.J. Geom. Phys. 62 (2012), 1489–1508. Zbl 1242.53046, MR 2911220, 10.1016/j.geomphys.2012.02.002 |
| Reference:
|
[9] Goertsches, O.: Riemannian supergeometry.Math. Z. 260 (2008), no. 3, 557–593. Zbl 1154.58001, MR 2434470 |
| Reference:
|
[10] Grabowski, J., Rotkiewicz, M.: Graded bundles and homogeneity structures.J. Geom. Phys. 62 (2011), 21–36. MR 2854191, 10.1016/j.geomphys.2011.09.004 |
| Reference:
|
[11] Hélein, F.: An introduction to supermanifolds and supersymmetry.Systèmes intégrables et théorie des champs quantiques (Baird, P., Hélein, F., Kouneiher, J., Pedit, F., Roubtsov, V., eds.), Hermann, 2009, pp. 103–157. |
| Reference:
|
[12] Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry.Springer–Verlag, 1993, http://www.emis.de/monographs/KSM/index.html. MR 1202431 |
| Reference:
|
[13] Lane, S. Mac: Categories for the working mathematician.Graduate Texts in Mathematics, vol. 5, Springer–Verlag, New York, 1998, Second Edition. MR 1712872 |
| Reference:
|
[14] Manin, Y.: Gauge Field Theory and Complex Geometry.Springer, 1997, Second Edition. Zbl 0884.53002, MR 1632008 |
| Reference:
|
[15] Rogers, A.: A global theory of supermanifolds.J. Math. Phys. 21 (1980), 1352–1365. Zbl 0447.58003, MR 0574696, 10.1063/1.524585 |
| Reference:
|
[16] Rogers, A.: Supermanifolds: theory and applications.World Scientific, 2007. Zbl 1135.58004, MR 2320438 |
| Reference:
|
[17] Shvarts, A.S.: On the definition of superspace.Teoret. Mat. Fiz. 60 (1984), no. 1, 37–42. Zbl 0575.58005, MR 0760438 |
| Reference:
|
[18] Van Proeyen, A.: Superconformal symmetry and higher-derivative Lagrangians.Breaking of supersymmetry and Ultraviolet Divergences in extended Supergravity, Frascati, 2013, arXiv:1306.2169 [hep-th]. |
| Reference:
|
[19] Varadarajan, V.S.: Supersymmetry for Mathematicians: An Introduction.Courant Lecture Notes, vol. 11, Providence, RI: American Mathematical Society, 2004. Zbl 1142.58009, MR 2069561 |
| Reference:
|
[20] Voronov, A.A.: Maps of supermanifolds.Teoret. Mat. Fiz. 60 (1984), no. 1, 43–48, (Russian). MR 0760439 |
| . |
