I am reading this paper and I found difficulty to develop the equation on the final of the page $3$, i.e.,
$$D_t V_a = \left\{ \frac{\partial V_k}{\partial t} + g^{ij} H H_{jk} V_i \right\} F_a^k.$$
I'm sure that the notation used in the paper is the same of my answer here because I could compute $D_a V_b$ as follows
\begin{align*} \nabla_a V_b &= (\nabla_a \omega_V)(F_b) = \nabla_a (\langle V, F_b \rangle) - \langle V, \nabla_a F_b \rangle = \langle \nabla_a V, F_b \rangle + \langle V, \nabla_a F_b \rangle - \langle V, \nabla_a F_b \rangle\\ &= \langle \nabla_a V, F_b \rangle = \langle \nabla_{F_a} V, F_b \rangle = \langle \nabla_{F_a^i \frac{\partial}{\partial x_i}} V, F_b^j \frac{\partial}{\partial x_j} \rangle = F_a^i F_b^j \left\langle \nabla_{\frac{\partial}{\partial x_i}} V, \frac{\partial}{\partial x_j} \right\rangle\\ &= F_a^i F_b^j \left( \frac{\partial}{\partial x_i} \left( \left\langle V, \frac{\partial}{\partial x_j} \right\rangle \right) - \left\langle V, \nabla_{\frac{\partial}{\partial x_i}} \frac{\partial}{\partial x_j} \right\rangle \right) = F_a^i F_b^j (\nabla_{\frac{\partial}{\partial x_i}} \omega_V) \left( \frac{\partial}{\partial x_j} \right)\\ &= F_a^i F_b^j \nabla_i V_b, \end{align*} where $\omega_X$ the $1$-form dual to a vector field $X$, i.e., $\omega_X(Y) := \langle X,Y \rangle$ for every vector field $Y$ on $M$ and fix the following notation: $\nabla_a X_b := (\nabla_a \omega_X) (F_b)$, but I couldn't compute $D_t V_a$ as you can read (see the evolution equation for $F_a^i$ and the notation of indexes on page $3$ of the paper for a better understanding of the computations below): \begin{align*} \frac{\partial V_a}{\partial t} &= (\nabla_{\frac{\partial}{\partial t}} \omega_V) (F_a) = \frac{\partial}{\partial t} \left( \langle V, F_a \rangle \right) - \langle V, \nabla_{\frac{\partial}{\partial t}} F_a \rangle\\ &= \left\langle \frac{\partial V}{\partial t}, F_a \right\rangle = \left\langle \frac{\partial }{\partial t} \left( \sum\limits_{b=1}^n \langle V, F_b \rangle F_b \right), F_a \right\rangle\\ &= \left\langle \left( \sum\limits_{b=1}^n \frac{\partial \langle V, F_b \rangle}{\partial t} F_b + \langle V, F_b \rangle \frac{\partial F_b}{\partial t} \right), F_a \right\rangle\\ &= \left\langle \left( \sum\limits_{b=1}^n \left( (\nabla_{\frac{\partial}{\partial t}} \omega_V) (F_b) + \langle V, \nabla_{\frac{\partial}{\partial t}} F_b \rangle \right) F_b + \langle V, F_b \rangle \frac{\partial F_b}{\partial t} \right), F_a \right\rangle\\ &= \left\langle \left( \sum\limits_{b=1}^n \left( (\nabla_{\frac{\partial}{\partial t}} \omega_V) (F_b) + \langle V, \nabla_{\frac{\partial}{\partial t}} F_b \rangle \right) F_b + \langle V, F_b \rangle g^{ij} H h_{kj} F_b^k \frac{\partial}{\partial x_i} \right), F_a \right\rangle\\ &= (\nabla_{\frac{\partial}{\partial t}} \omega_V) (F_a) + \left\langle \sum\limits_{b=1}^n \left( \langle V, \nabla_{\frac{\partial}{\partial t}} F_b \rangle \right) F_b + \langle V, F_b \rangle g^{ij} H h_{kj} F_b^k \frac{\partial}{\partial x_i}, F_a \right\rangle\\ &= (\nabla_{\frac{\partial}{\partial t}} \omega_V) (F_a^k \frac{\partial}{\partial x_k}) + \left\langle \sum\limits_{b=1}^n \left( \langle V, \nabla_{\frac{\partial}{\partial t}} F_b \rangle \right) F_b + \langle V, F_b \rangle g^{ij} H h_{kj} F_b^k \frac{\partial}{\partial x_i}, F_a \right\rangle\\ &= \frac{\partial V_k}{\partial t} F_a^k + \left\langle \sum\limits_{b=1}^n \left( \langle V, \nabla_{\frac{\partial}{\partial t}} F_b \rangle \right) F_b + \langle V, F_b \rangle g^{ij} H h_{kj} F_b^k \frac{\partial}{\partial x_i}, F_a \right\rangle \end{align*}
I would like to know how I can compute the evolution equation for $V_a$. Thanks in advance!