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Let $\Phi$ be a $C^{\infty }{ℝ}^n$-action on an orientable $(n+1)$-dimensional manifold. Assume $\Phi$ has an isolated compact orbit $T$ and let $W$ be a small tubular neighborhood of it. By a $C^{\infty }$ change of variables, we can write $W={ℝ}^n/{\mathbb{Z}}^n\times I$ and $T={𝕋}^n\times \left[0\right]$, where $I$ is some interval containing 0. In this work, we show that by a $C^0$ change of variables, $C^{\infty }$ outside $T$, we can make ${\Phi }_{|W}$ invariant by transformations of the type $(x,z)\to (x+a,z),a\in {ℝ}^n$, where $x\in {ℝ}^n/{\mathbb{Z}}^n$ and $z\in I$. As a corollary one cas describe completely the dynamics of $\Phi$ in $W$.

In this paper we give a geometric characterization of the 2-dimensional foliations on compact orientable 3-manifolds defined by a locally free smooth action of ${ℝ}^2$.

In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes,...