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



Isometric embedding of a neighbourhood of a totally real submanifold in a Kähler manifold
In general, there is no holomorphic isometric embedding of the desired kind. In fact, the Lagrangian $A$ is a bit of a red herring, because most real analytic Kähler metrics cannot, even locally, be holomorphically and isometrically embedded into $\mathbb{C}^N$ for any finite $N$. For example, see ["The complex version of Nash's Theorem is not true"](https://mathoverflow.net/questions/59738) for a discussion of why and some counterexamples.


