I'm reading this paper and I'm stuck in the lemma $3.2$ on page $5$. The author stated on the proof that
From [16] we have that
$$|\tilde{A}|^2 = |A|^2 + \kappa^2,$$
where the $\kappa = - \tilde{g}(\tilde{\nu},\overrightarrow{H}_{[x]})$ is the principal curvature of $\tilde{M}$ that belongs to the fiber direction.
I didn't understand where and how the reference $[16]$ gives me the equality, then I tried use the definition of the squared norm of $\tilde{A}$ to get the result. Let be $e_0$ a generator of the tangent fiber (defined on page $3$), then this vector normalized by the warped product metric $\tilde{g}$ (see page $24$ for the definition of $\tilde{g}$) is $e^{-f} e_0$, then
\begin{align*} |\tilde{A}|^2 &= \sum\limits_{i,j,k,l=0}^n \tilde{g}^{ij}\tilde{g}^{kl}\tilde{h}_{ik}\tilde{h}_{jl}\\ &= \tilde{g}^{00}\tilde{g}^{00}\tilde{h}_{00}\tilde{h}_{00} + \sum\limits_{i,j,k,l=1}^n \tilde{g}^{ij}\tilde{g}^{kl}\tilde{h}_{ik}\tilde{h}_{jl}\\ &= e^{-4f} \left( \tilde{g} \left( \tilde{\nabla}_{e^{-f}e_0} (e^{-f} e_0), \tilde{\nu} \right) \right)^2 + \sum\limits_{i,j,k,l=1}^n g^{ij}g^{kl}h_{ik}h_{jl}\\ &= e^{-4f} \left( \tilde{g} \left( \overrightarrow{H}_{[x]}, \tilde{\nu} \right) \right)^2 + |A|^2, \end{align*} where I used the proposition $2.2$ on page $3$, but I don't know what to do now. I will be grateful if someone can help me to obtain the relation between the squared norm of the second fundamental forms.