# Fubini Study Metric and Einstein constant

Hi all,

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?

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 $e_j$)

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Isn't this available in many different places, including Griffiths-Harris and wikipedia? –  Deane Yang Feb 15 '12 at 11:58
$$\mu=2\cdot n+3$$ ($\mathbb C\mathrm P^n$ is isometric to the factor $\mathbb S^{2n+1}/\mathbb S^1$. You can use O'Nail's formula to calculate sectional curvature, it is $=4$ in complex directions and $=1$ in real directions.) –  Anton Petrunin Feb 15 '12 at 14:29

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.
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.
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.
Your last sentence is not correct, the missing factor of $2$ come up when changing from real orthonormal frames to complex unitary frames. –  YangMills Feb 15 '12 at 15:30
typo: the metric should be $g_{i\bar{j}}$ –  John B Feb 16 '12 at 17:43