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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?


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Do you want the embedding of $U$ into $\mathbb{C}^N$ to be holomorphic as well as isometric? That requirement is not in your question, and the answer depends on whether you add it or not. – Robert Bryant Dec 7 '12 at 12:55
yes, is it possible if the embedding is holomorphic? – hapchiu Dec 7 '12 at 13:18
I thaught that one can do the following: since $A$ is real analytic we can use the real analytic version of Nash embedding theorem and consider $A$ as a real analytic Riemannian submanifold of some $\mathbb{R}^{N}$. Then locally $A$ is the zero set of some real analytic functions. Extend these functions holomorphically and then patch them together, since on the overlaps of some open sets in $A$ these real analytic functions are the same, and the extension of them would be the same holomorphic function. Is this possible? – hapchiu Dec 7 '12 at 13:29
up vote 2 down vote accepted

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" for a discussion of why and some counterexamples.

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