Secondary Chern-Euler forms and the law of vector fields (Q2880670)

In this paper the author gives two differential-geometric proofs of an equation called the Law of Vector Fields, which is a Poincaré-Hopf index theorem, stating that the total index of a vector field is equal to the Euler characteristic. More precisely, if \(V\) is a smooth vector field on a compact oriented Riemannian manifold \(X\) with boundary \(M\), then NEWLINE\[NEWLINE\mathrm{Ind}\, V + \mathrm{Ind}\, \partial_-V=\chi(X)NEWLINE\]NEWLINE where \(\partial_-V\) denotes the restriction of the projection of \(V|M\) onto \(TM\) along the normal direction to the part of \(M\) on which \(V\) is pointing inward. The key to the approach here is using a differential form \(\Phi\) on the tangent sphere bundle \(STX\) constructed in Chern's proof of the Gauss-Bonnet theorem such that it satisfies the relation \(d\Phi=-\pi^*\Omega\), where \(\pi^*\Omega\) is the pull back of the Euler curvature form \(\Omega\) of \(X\) (which is defined to be zero when \(\dim X\) is odd) to \(STX\) by the natural projection \(\pi : STX \to X\). This paper starts with the observation that proving the above formula is equivalent to proving the following: NEWLINE\[NEWLINE\int_{\overrightarrow{n}(M)}\Phi - \int_{\alpha_V(M)}\Phi = \mathrm{Ind}\, \partial_-VNEWLINE\]NEWLINE where \(\overrightarrow{n}\) denotes the outward unit normal vector field of \(M\) and \({\alpha_V(M)}\) the vector field \(V\) of \(M\) rescaled by setting \(\alpha_V(x)=V(x)/|V(x)|\) on \(M\) (here let \(V\) be assumed to be nowhere vanishing on \(M\), but there is no loss of generality in doing so). Thus this new formula becomes a focal point of this paper, so that the main part of the paper is devoted to the proof of this formula where it is carried out in two ways.