| Title:
|
A lossless reduction of geodesics on supermanifolds to non-graded differential geometry (English) |
| Author:
|
Garnier, Stéphane |
| Author:
|
Kalus, Matthias |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
50 |
| Issue:
|
4 |
| Year:
|
2014 |
| Pages:
|
205-218 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let ${\mathcal{M}}= (M,\mathcal{O}_\mathcal{M})$ be a smooth supermanifold with connection $\nabla $ and Batchelor model $\mathcal{O}_\mathcal{M}\cong \Gamma _{\Lambda E^\ast }$. From $({\mathcal{M}},\nabla )$ we construct a connection on the total space of the vector bundle $E\rightarrow {M}$. This reduction of $\nabla $ is well-defined independently of the isomorphism $\mathcal{O}_\mathcal{M} \cong \Gamma _{\Lambda E^\ast }$. It erases information, but however it turns out that the natural identification of supercurves in ${\mathcal{M}}$ (as maps from $ \mathbb{R}^{1|1}$ to $\mathcal{M}$) with curves in $E$ restricts to a 1 to 1 correspondence on geodesics. This bijection is induced by a natural identification of initial conditions for geodesics on ${\mathcal{M}}$, resp. $E$. Furthermore a Riemannian metric on $\mathcal{M}$ reduces to a symmetric bilinear form on the manifold $E$. Provided that the connection on ${\mathcal{M}}$ is compatible with the metric, resp. torsion free, the reduced connection on $E$ inherits these properties. For an odd metric, the reduction of a Levi-Civita connection on ${\mathcal{M}}$ turns out to be a Levi-Civita connection on $E$. (English) |
| Keyword:
|
supermanifolds |
| Keyword:
|
geodesics |
| Keyword:
|
Riemannian metrics |
| Keyword:
|
connections |
| MSC:
|
53B21 |
| MSC:
|
53C05 |
| MSC:
|
53C22 |
| MSC:
|
58A50 |
| idZBL:
|
Zbl 06487007 |
| idMR:
|
MR3291850 |
| DOI:
|
10.5817/AM2014-4-205 |
| . |
| Date available:
|
2014-12-09T14:40:20Z |
| Last updated:
|
2016-04-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144019 |
| . |
| Reference:
|
[1] Garnier, S., Wurzbacher, T.: The geodesic flow on a Riemannian supermanifold.J. Geom. Phys. 62 (6) (2012), 1489–1508. Zbl 1242.53046, MR 2911220, 10.1016/j.geomphys.2012.02.002 |
| Reference:
|
[2] Goertsches, O.: Riemannian supergeometry.Math. Z. 260 (3) (2008), 557–593. Zbl 1154.58001, MR 2434470 |
| Reference:
|
[3] Hohnhold, H., Kreck, M., Stolz, S., Teichner, P.: Differential forms and 0-dimensional supersymmetric field theories.Quantum Topol. 2 (1) (2011), 1–14. Zbl 1236.19008, MR 2763085, 10.4171/QT/12 |
| Reference:
|
[4] Michor, P.: Topics in Differential Geometry.American Mathematical Society, Providence, RI, 2008. Zbl 1175.53002, MR 2428390 |
| Reference:
|
[5] Monterde, J., Montesinos, A.: Integral curves of derivations.Ann. Global Anal. Geom. 6 (2) (1988), 177–189. Zbl 0632.58017, MR 0982764, 10.1007/BF00133038 |
| Reference:
|
[6] Schmitt, Th.: Super differential geometry.Tech. report, Report MATH, 84–5, Akademie der Wissenschaften der DDR, Institut für Mathematik, Berlin, 1984. Zbl 0587.58014, MR 0786297 |
| . |
