### A bumpy metric theorem and the Poisson relation for generic strictly convex domains.

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Let $\mathcal{M}=(M,{\mathcal{O}}_{\mathcal{M}})$ be a smooth supermanifold with connection $\nabla $ and Batchelor model ${\mathcal{O}}_{\mathcal{M}}\cong {\Gamma}_{\Lambda {E}^{*}}$. From $(\mathcal{M},\nabla )$ 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 ${\mathcal{O}}_{\mathcal{M}}\cong {\Gamma}_{\Lambda {E}^{*}}$. 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...

In the sub-Riemannian framework, we give geometric necessary and sufficient conditions for the existence of abnormal extremals of the Maximum Principle. We give relations between abnormality, ${C}^{1}$-rigidity and length minimizing. In particular, in the case of three dimensional manifolds we show that, if there exist abnormal extremals, generically, they are locally length minimizing and in the case of four dimensional manifolds we exhibit abnormal extremals which are not ${C}^{1}$-rigid and which can be minimizing...

We present an algorithm to generate a smooth curve interpolating a set of data on an $n$-dimensional ellipsoid, which is given in closed form. This is inspired by an algorithm based on a rolling and wrapping technique, described in [11] for data on a general manifold embedded in Euclidean space. Since the ellipsoid can be embedded in an Euclidean space, this algorithm can be implemented, at least theoretically. However, one of the basic steps of that algorithm consists in rolling the ellipsoid, over...