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.
> 
> 
> 



Factor of 2 In the Definition of Metric Contact Structure
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.


