# Show that $E_{t}=\left( \mathbf{x}_{t},\mathbf{E}_{1},\mathbf{E}_{2},\mathbf{E}_{3}\right)$ is a first order frame field along $\mathbf{x}_{t}$

Show that about any $m_{0}\in U$, there exist a neighborhood $V\subset U$ and a positive number $\delta \leq \epsilon$ for which there exist smooth vector fields $$\mathbf{E}_{a}:V\times \left( -\delta ,\delta \right) \rightarrow \mathbb{R}^3,$$ for $a=1,2,3$, such that $E_{t}=\left( \mathbf{x}_{t},\mathbf{E}_{1},\mathbf{% E}_{2},\mathbf{E}_{3}\right) :V\rightarrow E\left( 3\right)$ is a first order frame field along $x_{t}$ for each $t\in \left( -\delta ,\delta \right)$, with the property that $$\mathbf{E}_{1}\left( m,0\right) =\mathbf{e}_{1}\left( m\right) \text{, }% \mathbf{E}_{2}\left( m,0\right) =\mathbf{e}_{21}\left( m\right) \text{, }% \mathbf{E}_{3}\left( m,0\right) =\mathbf{e}_{3}\left( m\right)$$ for every $m\in V$. Consequently, $$\frac{\partial \mathbf{E}_{j}}{\partial t}\left( m,0\right) \cdot \mathbf{e}% _{j}\left( m\right) = \left. \frac{1}{2}\frac{\partial }{\partial t} \right|_{t=0}\left( \mathbf{E}_{j}\cdot \mathbf{E}_{j}\right) =0$$ for every $m\in V$ for $j=1,2$.

I only managed to show the existence of the fields, but I could not show the rest, that is, first order frame field and the others ...

We are using the definition below

Definition 8.1. An admissible variation of $\mathbf{x}$ is any smooth map $$X \colon M \times (-\epsilon, \epsilon) \to \mathbb{R}^3,$$ with compact support, such that for each $t \in (-\epsilon, \epsilon)$, the map $$\mathbf{x}_t \colon M \to \mathbb{R}^3, \quad \mathbf{x}_t(m) = X(m,t),$$ is an immersion. The support of $X$ is the closure in $M$ of the set of points of $M$ where $\mathbf{x}_t(m) \neq \mathbf{x}(m)$, for some $t$.

and

Definition 3.5 A first order frame field along $\mathbf{x}$ is a frame field $e \colon U \to G$ along $\mathbf{x}$ for which $$e^* \omega^{m+1} = \dotsb = e^* \omega^n = 0, \quad e^* \omega^1 \wedge \dotsb \wedge e^* \omega^m \neq 0.$$ at every point of $U$.

(Original scanned image here.)