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

Garnier, 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 -