We consider a Riemannian submersion π: M → N, where M is a CR-submanifold of a locally conformal Kaehler manifold L with the Lee form ω which is strongly non-Kaehler and N is an almost Hermitian manifold. First, we study some geometric structures of N and the relation between the holomorphic sectional curvatures of L and N. Next, we consider the leaves M of the foliation given by ω = 0 and give a necessary and sufficient condition for M to be a Sasakian manifold.

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...