I know this is a rather stupid question, but I still need to ask (and I am a physics student, so please excuse me using components):
In Blair's book and many other litereatures, the definition of a contact metric structure ($\kappa, R, g, \Phi$) contains a condition \begin{equation} g(X, \Phi Y) = d\kappa (X, Y). \end{equation}
Then I read Tanno's great paper Here on Generalized Tanaka-Webster connection, and several papers by others on various phy/math subjects, I see the definitions therein have a factor of $2$: \begin{equation} 2 g(X, \Phi Y) = d\kappa (X, Y) \end{equation}
To me, if I fix $\kappa$ as the contact 1-form, $d\kappa$ and $R$ are also fixed by $\kappa(R) = 1$. Then the condition $g(R, \cdot) = \kappa(\cdot)$ leaves NO rescaling freedom for $g$, which in the end leaves no rescaling freedom for $\Phi$.
So my question is:
Which one is correct, or both are actually reasonable?
Edit:
Actually, I find Blair's definition a bit strange: First, let $\nabla$ be the Levi-civita connection for a contact metric structure ($\kappa, R, g, \Phi$), then the defining equation is \begin{equation} g\left( {X,\Phi Y} \right) = d\kappa \left( {X,Y} \right) \Leftrightarrow {g_{mk}}{\Phi ^k}_n = \left[ {{\nabla _m}\left( {{\kappa _n}} \right) - {\nabla _n}\left( {{\kappa _m}} \right)} \right] \end{equation}
Also by Lemma 6.2, ${\nabla _X}R = - \Phi X - \Phi hX$, where $h \propto \mathcal{L}_R \Phi$.
Now, on a K-contact manifold, $h = 0$, and leaving \begin{equation} {\nabla _X}R = - \Phi X \Leftrightarrow {\nabla _m}{R^n} = - {\Phi ^n}_m \Leftrightarrow {\nabla _m}{\kappa_n} = - {\Phi _{nm}} = {\Phi _{mn}} \end{equation}
To me, this equation contradict the defining equation, by a factor of 2 (using Killing equation): \begin{equation} {\Phi _{mn}} = {\nabla _m}{\kappa _n} = \frac{1}{2}\left( {{\nabla _m}{\kappa _n} + {\nabla _m}{\kappa _n}} \right) = \frac{1}{2}\left( {{\nabla _m}{\kappa _n} - {\nabla _m}{\kappa _n}} \right) \end{equation}
