I want to know whether ot not Proposition 1.12 (2) in p. 120 of the book [KN] is typo.

[KN] : Foundations of differential geometry Volume II - Kobayashi and Nomizu 1969 Interscience Publishing

Let $\phi$ be a two form associated to Hermitian inner product $h$ such that $\phi(X,Y) = h(X,JY)$ where $J$ is an almost complex structure.

Then Proposition 1.12 (2) says that $\phi = -2i \sum_{j,k=1}^n h_{j\overline{k}} \xi^j \wedge \overline{\xi}^k$

I can not understand why 2 is needed in the above equation.

In the proof,
$\phi(Z,W)=-i\sum_{j,k=1}^n h_{j\overline{k}} (\xi^j(Z)\overline{\xi}^k(W)-\xi^j(W)\overline{\xi}^k (Z))$. This equation can be understood. In fact, in (5) of p. 155 the fundamental two form also has 2 (i.e. $\Phi = -2i \sum_{i, j} g_{ij} dz^i \wedge d\overline{z}^j$). Summarizingly, do Proposition 1.12 (2) in p. 120 and (5) of p. 155 have typos ?
Or, please give me an explanation.

  • 2
    $\begingroup$ @Hee Kwon Lee: Your question will become more readable if you enclose the TeX in dollar signs! Just click the edit button. If you have trouble, there is a link at the right of the page, below the list of related questions. $\endgroup$ – Mark Grant Apr 27 '12 at 14:11

Some authors define exterior products as quotient spaces of tensor products, letting $\alpha\wedge \beta$ be the coset represented by $\alpha\otimes\beta$. Others define exterior products as subspaces of tensor products, letting $\alpha\wedge \beta$ be $\alpha\otimes\beta-\beta\otimes\alpha$. Or one half of that. You can see that this can lead to confusion. This has been discussed elsewhere at MO.

I think you just have to make sure that you know what the convention is in that book.


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