Dans ce texte, on définit, pour les immersions lagrangiennes de variétés fermées dans ${ℂ}^n$, une notion d’aire symplectique enlacée. Puis on construit, dans le cas $n=2$, un certain nombre de surfaces lagrangiennes enlaçant une aire infinie. Dans le cas des surfaces exactes, elles ont le minimum de points doubles possible permis par la théorie (sauf la sphère), c’est-à-dire moins que prévu par quelques conjectures.

Let $F^{\mathrm{p}}\in \mathrm{GL}\left(3\right)$ be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form $\mathrm{Curl}\left[F^{\mathrm{p}}\right]·{\left(F^{\mathrm{p}}\right)}^T$ applied to rotations controls the gradient in the sense that pointwise $\forall R\in C^1({ℝ}^3,\mathrm{SO}\left(3\right)):\parallel \mathrm{Curl}\left[R\right]·R^T{\parallel }_{{𝕄}^{3\times 3}}^2\ge \frac{1}{2}{\parallel \mathrm{D}R\parallel }_{{ℝ}^{27}}^2$. This result complements rigidity results [Friesecke, James and Müller, Comme Pure Appl. Math. 55 (2002) 1461–1506; John, Comme Pure Appl. Math. 14 (1961) 391–413; Reshetnyak, Siberian Math. J. 8 (1967) 631–653)] as well as an associated linearized theorem...

Let $F^{\mathrm{p}}\in \mathrm{GL}\left(3\right)$ be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form $\mathrm{Curl}\left[F^{\mathrm{p}}\right]·{\left(F^{\mathrm{p}}\right)}^T$ applied to rotations controls the gradient in the sense that pointwise $\forall R\in C^1({ℝ}^3,\mathrm{SO}\left(3\right)):\parallel \mathrm{Curl}\left[R\right]·R^T{\parallel }_{{𝕄}^{3\times 3}}^2\ge \frac{1}{2}{\parallel \mathrm{D}R\parallel }_{{ℝ}^{27}}^2$. This result complements rigidity results [Friesecke, James and Müller, Comme Pure Appl. Math.55 (2002) 1461–1506; John, Comme Pure Appl. Math.14 (1961) 391–413; Reshetnyak, Siberian Math. J.8 (1967) 631–653)] as well as an associated linearized theorem saying...

These notes give an introduction to the equivalence problem of sub-Riemannian manifolds. We first introduce preliminaries in terms of connections, frame bundles and sub-Riemannian geometry. Then we arrive to the main aim of these notes, which is to give the description of the canonical grading and connection existing on sub-Riemann manifolds with constant symbol. These structures are exactly what is needed in order to determine if two manifolds are isometric. We give three concrete examples, which...

Conflict set are the points at equal distance from a number of manifolds. Known results on the differential geometry of these sets are generalized and extended.

In some recent work, fractal curvatures $C_k^f\left(F\right)$ and fractal curvature measures $C_k^f(F,·)$, $k=0,...,d$, have been determined for all self-similar sets $F$ in ${ℝ}^d$, for which the parallel neighborhoods satisfy a certain regularity condition and a certain rather technical curvature bound. The regularity condition is conjectured to be always satisfied, while the curvature bound has recently been shown to fail in some concrete examples. As a step towards a better understanding of its meaning, we discuss several equivalent formulations...

We investigate curvature properties of hypersurfaces in semi-Riemannian spaces of constant curvature with the minimal polynomial of the second fundamental tensor of second degree. We present suitable examples of hypersurfaces.