We prove Bertini type theorems for the inverse image, under a proper
morphism, of any Schubert variety in an homogeneous space. Using
generalisations of Deligne's trick, we deduce connectedness results for the
inverse image of the diagonal in $X^2$ where $X$ is any isotropic grassmannian.
We also deduce simple connectedness properties for subvarieties of $X$. Finally
we prove transplanting theorems {\`a} la Barth-Larsen for the Picard group of
any isotropic grassmannian of lines and for the Neron-Severi group of some
adjoint and coadjoint homogeneous spaces.