  We prove geometric rigidity inequalities for incompatible fields in dimension
higher than 2. We are able to obtain strong scaling-invariant $L^p$ estimates
in the supercritical regime, while for critical exponent $1^* = \frac{n}{n-1}$
we have a scaling invariant inequality only for the weak-$L^1$ norm. Although
not optimal, such an estimate in $L^{1 ,\infty}$ is enough in order to infer a
useful lemma which gives $BV$ bounds for $SO(n)$-valued fields with bounded
Curl.
