Let $M$ be a smooth manifold of 2n-dim, $v$ be a map from $M$ to the matrix of order $m\times m$. We call $p\in M$ is the singularity, if $v(p)$ is non-invertible. Suppose $v$ is smooth and the singularity is submanifold. Let $C$ be a connected component of singularity, $U$ is the tubular neighborhood of $C$. How to computing the integration $$\int_{\partial U}{\rm Tr}[(v^{-1}dv)^{2n-1}]$$

I guess its result is zero, may be that is wrong.