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 $t$.


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.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.