Fubini Study Metric and Einstein constant - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T04:42:22Z http://mathoverflow.net/feeds/question/88512 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88512/fubini-study-metric-and-einstein-constant Fubini Study Metric and Einstein constant Klaus Kröncke 2012-02-15T11:42:29Z 2012-02-16T19:26:17Z <p>Hi all,</p> <p>it is well known that the complex projective space with the fubini study metric is Einstein, but what is the explicit value, i.e. for which $\mu$ does $Ric=\mu g$ hold?</p> <p>Moreover, I would like to know how to calculate the sectional cuvature explicitly, because I would like to calculate the number $\sqrt{\sum K_{ij}}$ explicitly for a given orthonormal basis. ($K_{ij}$ is the sectional curvature of the plane spanned by $e_i$ and $e_j$)</p> http://mathoverflow.net/questions/88512/fubini-study-metric-and-einstein-constant/88525#88525 Answer by Renato G Bettiol for Fubini Study Metric and Einstein constant Renato G Bettiol 2012-02-15T15:20:59Z 2012-02-16T19:26:17Z <p>As suggested by Anton, you can use the O'Neill formulas in the Riemannian submersion $\mathbb C^{n+1}\to \mathbb{C} P^n$ that defines the Fubini-Study metric on $\mathbb C P^n$. This gives the following: suppose $X,Y$ are orthonormal tangent vectors at some point in $\mathbb C P^n$, and denote by $\overline X,\overline Y$ their horizontal lifts to $\mathbb C^{n+1}$ (which are also orthonormal). Then $$sec(X,Y)=1+\tfrac34\|[\overline X,\overline Y]^v\|^2=1+3|\overline g(\overline Y,J\overline X)|^2,$$ where $\overline g$ is the canonical Euclidean metric on $\mathbb C^{n+1}$, $()^v$ denotes the vertical component wrt the submersion and $J$ is the complex structure, i.e., multiplication by $\sqrt{-1}$. Note that this immediately implies that $\mathbb CP^n$ is $\tfrac14$-pinched.</p> <blockquote> <p>With the above formula, you can easily compute the Einstein constant of $\mathbb C P^n$ to be equal to $\mu=2n+2$, see e.g. Petersen's book "Riemannian Geometry", chapter 3.</p> </blockquote> <p>Another possible way of doing it is using that this is a Kahler manifold. The Fubini-Study metric can be thought of as $\omega_{FS}=\sqrt{-1}\partial\overline\partial\log\|z\|^2$, where $\|z\|^2$ is the square norm of a local non vanishing holomorphic section (it is independent of the choice of section by the $\partial\overline\partial$-lemma). You can then compute in local normal (holomorphic) coordinates the coefficients $g_{i\bar j}$ and use that the Ricci form is given by $Ric(\omega)=-\sqrt{-1}\partial\overline\partial\log\det(g_{i\bar{j}})$. This will obviously give you the same result, but in the form $Ric(\omega_{FS})=(n+1)\omega_{FS}$. As pointed out in the comments below, the reason for the missing factor $2$ in this computation is that we have to change from real orthonormal frames to complex unitary frames.</p>