calabi yau and ricci flat metrics - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-20T16:23:14Z http://mathoverflow.net/feeds/question/107901 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107901/calabi-yau-and-ricci-flat-metrics calabi yau and ricci flat metrics bruno 2012-09-23T13:49:05Z 2012-09-24T12:22:59Z <p>hallo,</p> <p>I have the following question: Let $X$ be a Calabi-Yau manifold ($X$ might be compact with or without boundary). How can one prove then that the metric is Ricci flat? I would be very thankfull for answers. My definition of calabi-yau manifolds is: $(X,\omega, \Omega)$ is Calabi-Yau if we have $\frac{\omega^{n}}{n!} = (-1)^{\frac{n(n-1)}{2}} (\frac{i}{2})^{n} \Omega \wedge \bar{\Omega}$, where $\Omega$ is the holomorphic volume form and $\omega$ is the Kähler form. I am wondering if one can show that $\omega$ is then Ricci-flat? </p> <p>bruno</p>