A multidimensional generalization of the Gauss–Bonnet formula for vector fields in Euclidean space

A unit vector field $n$ is considered, defined on some neighborhood $G$ in $(m+1)$-dimensional Euclidean space $E^{m+1}$, for which a formula is established that generalizes the Gauss–Bonnet formula. For this purpose, using the vector field $n$, a map is constructed from an arbitrary hypersurface $F^m\subset G$ onto the $m$-dimensional unit sphere $S^m$. It is proved that the volume element $d\sigma$ of the sphere $S^m$ and the volume element $dV$ of the hypersurface $F^m$ are connected under this map by the relation $d\sigma=(P\nu)dV$, where $\nu$ is the unit normal to $F^m$ and $P$ is a vector of the curvature of the field $n$:
$$ P=(-1)^m\{S_mn+S_{m-1}k_1+\dots+k_m\}. $$
Here the $S_i$ are symmetric functions of the principal curvatures of the second kind of the field $n$, $k_1=\nabla_nn,\dots,k_{i+1}=\nabla_{k_i}n,\dots$. The flux of the vector field $P$ through a closed hypersurface $F^m$, divided by the volume of the $m$-dimensional unit sphere $S^m$, equals the degree of the map of $F^m$ to $S^m$ determined by the vector field $n$. For a field $n$, given on all of $E^3$, including the point at infinity, the Hopf invariant is calculated by use of the vector field $P$.
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