Hallo,

Let $(M,J,\omega)$ be a real-analytic Kähler manifold. Let furthermore $A \subset M$ be a real analytic, totally real, Lagrangian submanifold and set $g := h|_{A}$. Where $h$ is the Kähler metric on $M$. $g$ is now a Riemannian metric on $A$. Let $U$ be an arbitrary small neigbourhood of $A$ in $M$. Is it possible to embedd $U$ in some $\mathbb{C}^{N}$ isometrically? I think its always possible to embed such an arbitrary small neighbourhood $U$ in $\mathbb{C}^{N}$ for some $N$. But can this be also done isometrically?

hapchiu