A multidimensional generalization of the Gauss–Bonnet formula for vector fields in Euclidean space
Yu. A. Aminov
Abstract:
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
$P$.
Bibliography: 5 titles.
UDC:
514
MSC: Primary
53A07; Secondary
55Q25 Received: 21.10.1986
Fulltext:
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