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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...

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