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Non-split almost complex and non-split Riemannian supermanifolds

Matthias Kalus — 2019

Archivum Mathematicum

Non-split almost complex supermanifolds and non-split Riemannian supermanifolds are studied. The first obstacle for a splitting is parametrized by group orbits on an infinite dimensional vector space. For almost complex structures, the existence of a splitting is equivalent to the existence of local coordinates in which the almost complex structure can be represented by a purely numerical matrix, i.e. containing no Grassmann variables. For Riemannian metrics, terms up to degree 2 are allowed in...

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

Stéphane GarnierMatthias Kalus — 2014

Archivum Mathematicum

Let = ( M , 𝒪 ) be a smooth supermanifold with connection and Batchelor model 𝒪 Γ Λ E * . From ( , ) we construct a connection on the total space of the vector bundle E M . This reduction of is well-defined independently of the isomorphism 𝒪 Γ Λ E * . It erases information, but however it turns out that the natural identification of supercurves in (as maps from 1 | 1 to ) 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...

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