  We consider the following question: Given a connected open domain
$\Omega\subset R^n$, suppose $u,v:\Omega\rightarrow R^n$ with $\det(\nabla
u)>0$, $\det(\nabla v)>0$ a.e. are such that $\nabla u^T(x)\nabla u(x)=\nabla
v(x)^T \nabla v(x)$ a.e. does this imply a global relation of the form $\nabla
v(x)= R\nabla u(x)$ a.e. in $\Omega$ where $R\in SO(n)$? If $u,v$ are $C^1$ it
is an exercise to see this true, if $u,v\in W^{1,1}$ we show this is false. We
prove this question has a positive answer if $v\in W^{1,1}$ and $u\in W^{1,n}$
is a mapping of $L^p$ integrable dilatation for $p>n-1$. These conditions are
sharp in two dimensions and this result represents a generalization of the
corollary to Liouville's theorem that states that the differential inclusion
$\nabla u\in SO(n)$ can only be satisfied by an affine mapping.
  Liouville's corollary for rotations has been generalized by Reshetnyak who
proved convergence of gradients to a fixed rotation for any weakly converging
sequence $v_k\in W^{1,1}$ for which $$ \int_{\Omega} \mathrm{dist}(\nabla
v_k,SO(n)) dz\rightarrow 0 \text{as} k\rightarrow \infty. $$ Let $S(\cdot)$
denote the (multiplicative) symmetric part of a matrix. In Theorem 3 we prove
an analogous for any pair of weakly converging sequences $v_k\in W^{1,p}$ and
$u_k\in W^{1,\frac{p(n-1)}{p-1}}$ (where $p\in \left[1,n\right]$ and the
sequence $(u_k)$ has its dilatation pointwise bounded above by an $L^r$
integrable function, $r>n-1$) that satisfy $\int_{\Omega} \left|S(\nabla
u_k)-S(\nabla v_k)\right|^p dz\rightarrow 0$ as $k\rightarrow \infty$ and for
which the sign of the $\det(\nabla v_k)$ tends to 1 in $L^1$. This result
contains Reshetnyak's theorem as the special case $(u_k)\equiv Id$, p=1.
