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In Blair's book and many many literatures, I see definition of a contact metric manifold which involves a relation \begin{equation} d\kappa \left( {X,Y} \right) = g\left( {X,\Phi Y} \right) \end{equation} as well as other standard relation like ${\Phi ^2} = - 1 + R \otimes \kappa $, $\Phi R = 0$, etc.

On the other hand, in Tanno's papers and a few others literatures, the definitions involves \begin{equation} d\kappa \left( {X,Y} \right) = 2 g\left( {X,\Phi Y} \right) \end{equation} differing from Blair's by a factor of $2$.

So I try to figure out which is "more correct", and then comes one important equation that really confuses me, in Blair's book (Lemma 6.2): \begin{equation} {\nabla _X}R = - \Phi X - \Phi hX, \;\;\;\;h \propto \mathcal{L}_R \Phi\;\;\;\;\;\;\;\;(*) \end{equation}

Now, let us consider the contact metric structure to be K-contact, and therefore $h = 0$, ${\mathcal{L}_R}g = 0$, and also equation $(*)$ should still hold, hence (in components) \begin{equation} {\nabla _X}R = - \Phi X \Leftrightarrow {\nabla _m}{R^n} = - {\Phi ^n}_m \Rightarrow {\nabla _m}{R_n} = - {g_{nk}}{\Phi ^k}_m\;\;\;\;\;\;\;\;(**) \end{equation}

On the other hand, since $\nabla$ is Levi-civita, we have \begin{equation} {d\kappa } \left( {X,Y} \right) = \left( {{\nabla _m}{R_n} - {\nabla _n}{R_m}} \right){X^m}{Y^n} = g\left( {X,\Phi Y} \right) = {g_{mk}}{\Phi ^k}_n{X^m}{Y^n} \end{equation} and therefore ${\nabla _m}{R_n} - {\nabla _n}{R_m} = {g_{mk}}{\Phi ^k}_n$. Now comes the final point: using the Killing vector equation ${\nabla _m}{R_n} = - {\nabla _n}{R_m}$, one obtains \begin{equation} - 2{\nabla _m}{R_n} = 2{\nabla _n}{R_m} = - {g_{mk}}{\Phi ^k}_n \end{equation} which directly contradict the equation $(**)$

So my question is

Is Blair's definition or Tanno's "more correct"? Is Blair's definition incompatible with the Lemma 6.2? Note that Tannos definition has the correct factor of 2, so $(*)$ is compatible with the definition.

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The discrepancy comes from which coefficient convention is being used for the covariant derivative d. Blair uses the convention that for any 1-form $\eta$ $$d\eta(X,Y) = \frac{1}{2}\left( X\eta(Y) - Y\eta(X) - \eta[X,Y]\right)$$ whereas Tanno is most likely using $$d\eta(X,Y) = X\eta(Y) - Y\eta(X) - \eta[X,Y].$$

The difference between these is conceptually nil, but computationally important. And, in fact, it is derived from the wedge product convention for 1-forms: $$\eta \wedge \zeta = \frac{1}{2}(\eta \otimes \zeta - \zeta \otimes \eta)$$ versus $$\eta \wedge \zeta = \eta \otimes \zeta - \zeta \otimes \eta.$$

Neither is more correct than the other. It's just which convention one chooses.

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  • $\begingroup$ I agree with Brendan on the two definitions of $d$: I found at the bottom of page 62, Blair wrote down $d\eta$ formula. I feel that a clear list of convention should be provided at the very beginning of the book. Thanks! $\endgroup$ – Lelouch May 4 '14 at 3:33

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