I have the following situation, $N$ is a $k$-manifold, $X$ a compact $(k+1)$-manifold and $F:X \to N$ a smooth map. Let $y$ be a regular value of both $F$ and $F|_{\partial X}$, hence $F^{-1}(y)$ is a compact $1$-manifold. Let $J$ be a connected component of $\partial X\cap f^{-1}(y)$ having a nonempty border, $c:[0,1] \to J$ is a diffeomorphism, $\{w_1,\dots,w_k\}$ a basis of $T_yN$ and $\{v_1^j,\dots,v_k^j\}$ a basis of $T_{c(j)}\partial X$ for $j = 0,1$ such that $dF_{c(j)} v_i^j = w_i$. I'm supposed to build vector fields $V_i(t) = (c(t),v_i(t))$ over $c$ such that $dF_{c(t)} v_i(t) = w_i$ and $v_i(j) = v_i^j$, for $j = 0,1$.
The hint is to obtain a contradiction if the supremum of $$\{s \in [0,1]\mid \text{ there exist vector fields $V_i$ over $c|_{[0,s]}$ satisfying the desired conditions}\}$$ is less than $1$, but it did not help me. I suspect that I have to use some result about extending curves $v_i:[0,s]\to T\partial X$ under the restriction $dF_{c(t)}v_i(t) = w_i$, but I haven't found anything about it. Is this the correct way? Thanks, guys.
I think that "Let $J$ be a connected component of $\partial X$" should this be $\partial X\cap f^{-1}(y)$. Otherwise it's not clear why $c(t)$ projects to $y$ under $F$.
The idea of the exercise is that being a submersion, $F$ can be locally represented by projection $(x_0,\dots,x_k)\mapsto (x_1,\dots,x_k)$ in appropriate local coordinates. The parameter $x_0$ parametrizes the curve. In these coordinates an extension of $V_i$ is obtained by letting it be independent of $x_0$; that is, expand $v_i(t_0)$ as $\sum c_j\frac{\partial}{\partial x_j}$ at $c(t_0)$ and then define $v_i(t)=\sum c_j\frac{\partial}{\partial x_j}$ at neighboring points of the curve.
The above construction should be done in a neighborhood containing $c(s)$ where $s$ is defined in the hint. It provides an extension of the fields beyond $s$, proving that the assumption $s<1$ yields a contradiction.