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 $e_j$)

  • 1
    $\begingroup$ Isn't this available in many different places, including Griffiths-Harris and wikipedia? $\endgroup$ – Deane Yang Feb 15 '12 at 11:58
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    $\begingroup$ $$\mu=2\cdot n+3$$ ($\mathbb C\mathrm P^n$ 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.) $\endgroup$ – 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.

  • $\begingroup$ Your last sentence is not correct, the missing factor of $2$ come up when changing from real orthonormal frames to complex unitary frames. $\endgroup$ – YangMills Feb 15 '12 at 15:30
  • $\begingroup$ @YangMills: thank you! I just corrected it! $\endgroup$ – Renato G. Bettiol Feb 15 '12 at 17:04
  • $\begingroup$ typo: the metric should be $g_{i\bar{j}}$ $\endgroup$ – J. GE Feb 16 '12 at 17:43
  • $\begingroup$ @MG: Thanks! I guess this fixes it. In the code I actually had typed i\overline j, but I agree the result seemed more like \overline{ij}. Using \bar instead seems to give a better output... thanks! $\endgroup$ – Renato G. Bettiol Feb 16 '12 at 19:28

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