We know that Kahler curvature tensor can be decomposed into three items:scalar part, traceless Ricci part and Bochner curvature tensor. In page 77 of Besse's book, it appears two symbols: $B$ and $B_0$. I don't know which one stands for exactly the Bochner curvature tensor. If someone is Bochner tensor curvature, then what is the meaning of the other one? I know the local complex coordinate's expression for Bocher curvature tensor ($B_{i\bar{j}k\bar{l}}=\ldots$). What is the concrete local coordinate expression for the other one?

Also, I did some calculations myself and found that some of them don't agree with those of Besse's formula (2.67). For example, I think $s^2=4m(m-1)|U|^2$ should be $s^2=m(m-1)|U|^2$. Since I don't know the concrete local complex coordinate expression of $B_0$, so I cannot check the first formula in (2.67).

Thanks in advance.


Well, $B$, as defined in Besse on page 77, is not the Bochner curvature because it is not traceless when $m>1$ (cf. (2.64)). I believe that $B_0$ is some version of the Bochner curvature tensor.

Incidentally, the decomposition can be understood a little better by realizing that the space of curvature tensors of Kähler metrics at a point $p\in M$ is isomorphic to $P_{(2,2)}$ the space of real-valued polynomials on the complex vector space $T_pM$ that are homogeneous of bi-degree $(2,2)$. (Essentially, it's the degree $(2,2)$ part of the Taylor series of a Kähler potential function for the metric.) The metric itself at $p$ can be regarded as a positive definite real polynomial on $T_pM$ that is homogeneous of bi-degree $(1,1)$. Given $g_p\in {P^+}_{(1,1)}$, the space $P_{(2,2)}$ decomposes into subspaces as $$ P_{(2,2)} = P^0_{(2,2)} \oplus \bigl(P^0_{(1,1)}\cdot g_p\bigr) \oplus \bigl(\mathbb{R}\cdot (g_p)^2\bigr), $$ where $P^0_{(k,k)}$ is the space of real-valued polynomials on $T_pM$ that are homogeneous of bi-degree $(k,k)$ and traceless with respect to $g_p$. This is the decomposition into the Bochner curvature, the traceless Ricci curvature, and the scalar curvature.

As for the normalizations in the formulas, you have to be careful with Besse because, in different parts of the book, different norms (all differing by constant scalar multiples, of course) are taken on the various tensor fields. For example, in some places, Besse takes the $2$-form $dx^i\wedge dx^j$ (on standard $n$-space) to have norm $1$, but, in other places, where it is regarded as $dx^i\otimes dx^j - dx^j\otimes dx^i$ embedded in the tensor product and thereby inheriting the 'natural' inner product, it is regarded as having norm $\sqrt2$. Make sure your normalization in your formulas agrees with Besse's when you are checking your constants.

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  • $\begingroup$ Dear Professor Bryant, according to your 2001 JAMS's paper and Kamishima's results, the only simply-connected comapct Bochner-Kahler manifolds are $CP^n$, isn't it? $\endgroup$ – Ping May 10 '12 at 2:38
  • $\begingroup$ @Ping Li: That's correct. $\endgroup$ – Robert Bryant May 14 '12 at 15:26
  • $\begingroup$ @Robert Bryant: Dear Professor Bryant, all $M^p_c\times M^{n-p}_{-c}$ have constant scalar curvature (CSC), so their compact quotients are still CSC. Is this correct? If so, then that means B-K implies CSC. Am I right? $\endgroup$ – Ping May 15 '12 at 0:17
  • $\begingroup$ @Ping Li: B-K does not imply CSC except in the compact case. There are complete B-K metrics on $\mathbb{C}^n$ that are not CSC. $\endgroup$ – Robert Bryant May 15 '12 at 0:33
  • $\begingroup$ @Robert Bryant: Thanks very much. What I mean is just the compact case. I know that in the noncompact case your paper corrected Professor Kamishima's original mistake. $\endgroup$ – Ping May 15 '12 at 1:34

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