Assume $e$ and $Q$ are smooth $n-1$ dimensional submanifolds in $\mathbb{R}^n$ and $X:Q\rightarrow e$ is a diffeomorphism with $c\leq\vert\det(JX)\vert\leq C$, where $JX$ is the Jacobian matrix of $X$. Let $\gamma\subset e $ be a smooth curve.
I want to show $\vert\gamma\vert\leq C\vert X^{-1}(\gamma)\vert$ where $\vert\cdot\vert$ is the $1$ dim. Hausdorff measure.
Basically I just need $$\det(J_pF^TJ_{F(p) } X^TJ_{F(p)}XJ_pF)\leq \det(J_{F(p) }X^TJ_{F(p) }X)\cdot \det(J_pF^TJ_pF),$$ with $F=X^{-1}\circ\gamma$. But I'm not sure if this is even true, since $J(X^{-1}\circ\gamma)$ isn't a square matrix.
EDIT: Add that $e$ and $Q$ are compact and X a $C^\infty$-diffeomorphism .
The bound on $\det (JX)$ is no good in controlling the lower dimensional objects. For example, let $n\ge 3$, $Q = e = \mathbb R^{n-1} \times \{0\} \subset \mathbb R^n$ and for any $t>0$, let $X: Q\to e$ be the linear map given by the matrix
$$ X = \begin{bmatrix} t & 0 & \\ 0 & t^{-1} & \\ & & I_{n-3} \end{bmatrix}$$
Then $JX = X$ has $\det (JX) = 1$. If $\gamma(s) = (s, 0,\cdots, 0)$ where $s\in [-1, 1]$, then $$X^{-1} \gamma = \{ (s, 0, \cdots, 0) : s\in [-t^{-1}, t^{-1}]\}.$$ Then $|X^{-1} \gamma|$ can be as small as we like by choosing $t\to \infty$.
What you need is the bound on the whole $JX$, or put it differently, bounds on the Lipschitz constant of the map $X$.