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    *Francesco Fanelli*\
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### Abstract {#abstract .unnumbered}

[We obtain a result about propagation of geometric properties for solutions of the non-homogeneous incompressible Euler system in any dimension $N\geq2$. In particular, we investigate conservation of striated and conormal regularity, which is a natural way of generalizing the $2$-D structure of vortex patches. The results we get are only local in time, even in the dimension $N=2$; however, we provide an explicit lower bound for the lifespan of the solution. In the case of physical dimension $N=2$ or $3$, we investigate also propagation of Hölder regularity in the interior of a bounded domain.]{}

Introduction
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In this paper we are interested in studying conservation of geometric properties for solutions of the density-dependent incompressible Euler system $$\label{eq:ddeuler}
\left\{\begin{array}{l}
\d_t\rho\,+\,u\cdot\nabla \rho\,=\,0\\[1ex]
\rho\left(\d_tu\,+\,u\cdot\nabla u\right)\,+\,\nabla\Pi\,=\,0\\[1ex]
{{\rm div}\,}u\,=\,0\,,
\end{array}\right.$$ which describes the evolution of a non-homogeneous inviscid fluid with no body force acting on it, an assumption we will make throughout all this paper to simplify the presentation. Here, $\rho(t,x)\in{\mathbb{R}}_+$ represents the density of the fluid, $u(t,x)\in{\mathbb{R}}^N$ its velocity field and $\Pi(t,x)\in{\mathbb{R}}$ its pressure. The term $\nabla\Pi$ can be also seen as the Lagrangian multiplier associated to 