# Relations of Kahler manifold ; Einstein manifold ; Kahler-Einstein manifold ?

If $L$ is a closed n-dimensional manifold, smoothly immersed into a $2n$-dimensional Kähler manifold $M$. Denote the complex structure on $M$ by $J$.

Now we choose normal coordinates for both $L$ and $M$, such that $\{e_{i}\},i=1,\cdots, n$ are the basis of tangent space on $L$, $\{Je_{i}\},i=1,\cdots, n$ are the basis of normal space on $L$.

Here $\overline{R}ic$ is the Ricci tensor on $M$.

Is it possible that $\overline{R}ic(Je_{i},e_{j})=0, \forall i,j=1,\cdots, n$ ?

Note here that we require $M$ is not Einstein. Can you give an example?

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Please consider editing your question: (i) the tags should be subject-specific - this question is not about math communication except in a sense that applies to every question on the site. (ii) Are you are assuming the immersion totally real? (iii) The phrasing of the question isn't clear to me. If you are looking for a non Einstein example, this is easy - take your favourite Kaehler-Einstein example, then deform the Kaehler metric away from $L$. If that's not what you want, please try to phrase your question more precisely. –  Tim Perutz Jun 7 '10 at 16:45
You can't assume that the $Je_i$'s span the normal space unless $L$ is a Lagrangian submanifold of $M$ with respect to the symplectic structure induced from the K\"ahler form. –  Spiro Karigiannis Sep 6 '10 at 23:44