Geodesic
A lossless reduction of geodesics on supermanifolds to non-graded differential geometry
Archivum mathematicum, Tome 50 (2014) no. 4, pp. 205-218
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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 $E$.
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 $E$.
DOI : 10.5817/AM2014-4-205
Classification : 53B21, 53C05, 53C22, 58A50
Keywords: supermanifolds; geodesics; Riemannian metrics; connections 
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Garnier, Stéphane; Kalus, Matthias. A lossless reduction of geodesics on supermanifolds to non-graded  differential geometry. Archivum mathematicum, Tome 50 (2014) no. 4, pp. 205-218. doi: 10.5817/AM2014-4-205

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