On Sha's secondary Chern-Euler class (Q2909644)

Let \(X\) be a compact connected oriented smooth \(n\)-dimensional manifold with boundary \(M\), where \(n\geq 2\). Fix a Riemannian metric on \(X\). Denote by \(STX\) the unit sphere bundle associated to the tangent bundle of \(X\). Let \(n:M\to STX|_M\) denote the unit outward pointing normal normal vector field on \(M\). Let \(CTM\) be the complement of \(Im(n)\cup Im(-n)\) in \(STX|_M\) so that \(CTM\to M\) is a bundle whose fibre over \(x\) is the cylinder \(ST_xX\setminus\{n(x),-n(x)\}\cong \mathbb{S}^{n-1}\times (-1,1)\).NEWLINENEWLINEChern constructed an \((n-1)\)-form \(\Phi\) on the unit sphere bundle \(STX\) which is closed when \(n\) is odd, and, when \(n\) is even, \(d\Phi=-\Omega\) where \(\Omega\) is the pull-back to \(STX\) of the Euler curvature form of \(X\). In any case \(d\Phi|_M=0\) for dimension reasons.NEWLINENEWLINELet \(V\) be a smooth vector on \(X\) with at most finite many zeros contained in the interior of \(X\). Let \(\alpha_V:M\to STX|_M\) be defined as \(x\to V(x)/||V(x)||\). Assuming that the Riemannian metric restricts to a product metric in a collar neighbourhood \(U\cong M\times [0,1)\) of \(M\subset X\), \textit{J.-P. Sha} [Ann. Math. (2) 150, No. 3, 1151--1158 (1999; Zbl 0980.57012)] obtained the relative Euler-Poincaré formula that \(Ind(V)-\int_{\alpha_V(M)}\Phi|_{\alpha_V(M)}\) equals \(0\) if \(\dim X\) is odd and equals \(\chi(X)\), the Euler-Poincaré characteristic of \(X\), if \(\dim(X)\) is even. Under the same condition on the Riemannian metric, the author shows that \(\Phi|_{CTM}\) is exact. For `generic' \(V\), the author computes \(\int_{\alpha_V(M)}\Phi\) in terms of the indices of \(\partial_-V, \partial_+V\) where \(\partial V\) is the projection of \(V|_M\) to \(TM\) and \(\partial_+V\) and \(\partial_-V\) are restrictions of \(\partial V\) to the subspaces of \(M\) where \(V\) points inwards and outwards respectively. This leads to a new proof of the `law of vector fields', namely, \(Ind(V)+Ind(\partial_-V)=\chi(X)\).