# A lossless reduction of geodesics on supermanifolds to non-graded differential geometry

Stéphane Garnier; Matthias Kalus

Archivum Mathematicum (2014)

- Volume: 050, Issue: 4, page 205-218
- ISSN: 0044-8753

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topGarnier, Stéphane, and Kalus, Matthias. "A lossless reduction of geodesics on supermanifolds to non-graded differential geometry." Archivum Mathematicum 050.4 (2014): 205-218. <http://eudml.org/doc/262126>.

@article{Garnier2014,

abstract = {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 $E$.},

author = {Garnier, Stéphane, Kalus, Matthias},

journal = {Archivum Mathematicum},

keywords = {supermanifolds; geodesics; Riemannian metrics; connections; supermanifolds; geodesics; Riemannian metrics; connections},

language = {eng},

number = {4},

pages = {205-218},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {A lossless reduction of geodesics on supermanifolds to non-graded differential geometry},

url = {http://eudml.org/doc/262126},

volume = {050},

year = {2014},

}

TY - JOUR

AU - Garnier, Stéphane

AU - Kalus, Matthias

TI - A lossless reduction of geodesics on supermanifolds to non-graded differential geometry

JO - Archivum Mathematicum

PY - 2014

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 050

IS - 4

SP - 205

EP - 218

AB - 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 $E$.

LA - eng

KW - supermanifolds; geodesics; Riemannian metrics; connections; supermanifolds; geodesics; Riemannian metrics; connections

UR - http://eudml.org/doc/262126

ER -

## References

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- Michor, P., Topics in Differential Geometry, American Mathematical Society, Providence, RI, 2008. (2008) Zbl1175.53002MR2428390
- Monterde, J., Montesinos, A., 10.1007/BF00133038, Ann. Global Anal. Geom. 6 (2) (1988), 177–189. (1988) Zbl0632.58017MR0982764DOI10.1007/BF00133038
- Schmitt, Th., Super differential geometry, Tech. report, Report MATH, 84–5, Akademie der Wissenschaften der DDR, Institut für Mathematik, Berlin, 1984. (1984) Zbl0587.58014MR0786297

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