Hallo,


It is a known fact that any real-analytic Riemannian manifold $M$ admits a isometric embedding in a Kähler manifold $\Omega$, where $M$ is totally real in $\Omega$. Of $\Omega$ can be taught of as some open neighbourhood of the zero section of the cotangent bundle $T^{\*}M$. This complex manifold is far from being compact. My question is now: Can one choose the complex manifold $\Omega$ not to be just some open neighbourhood of $M$ but to be a compact Kähler manifold? In other words: Can one embed any real-analytic compact Riemannian manifold isometrically in a compact Kähler manifold, such that the real-analytic Riemannian manifold is totally real in the Kähler manifold? Are there any references on this topic? What furter assumptions does one need in order that this works? Or is it entirely impossible? If so, why?
By compact I mean compact without boundary!


hapchiu



Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold
I think the answer to your question should be positive and below is a sketch of what should work (I think).


Any real analytic manifold can be realised as the real part of a complex projective manifold. I.e. one should embedd the real analytic manifold smoothly in $\mathbb R^n$ and then approximate by a real algebraic submanifold, whose complexification will be the Kahler manifold you are thinking about. 
Now, the emdedding can be done almost isometrically. I think, using this one should be able to show that the standard (say Fubini-Study) metric on the complexification can be adjusted a bit to make the embedding in the complexification an isometry.


