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