The secondary Chern-Euler class for a general submanifold (Q2888791)

Let \(X\) be a compact connected oriented smooth Riemannian manifold of dimension \(n\). Suppose that \(M\hookrightarrow X\) is a codimension \(m\) submanifold of \(X\) which is closed and connected. \textit{S.-S. Chern} defined in [Ann. Math. (2) 46, 674--684 (1945; Zbl 0060.38104)] an \((n-1)\)-form \(\Phi\) on the unit sphere bundle \(STX\) associated to the tangent bundle \(TX\) of \(X\) and proved the formula \(\int_N \Phi|_N=\chi(M)\), \(\chi\) being the Euler-Poincaré characteristic of \(M,\) where \(N\subset STX\) denotes the unit sphere bundle of the normal bundle to the imbedding \(M\subset X\).NEWLINENEWLINEIt turns out that the restriction of \(\Phi\) to \(STX|_M\) is closed and hence defines a cohomology class \([\Phi]\in H^{n-1}(STX|_M;\mathbb{R})\). This is the \textit{secondary Chern-Euler} class of the title. The author proves in Theorem 2.1 that \([\Phi]\) is an integral class independent of the Riemannian metric provided \(m\geq 2\); when \(m=1\) and \(n\) odd, \(2[\Phi]\) is integral and independent of the Riemannian metric. Otherwise, the class depends on the Riemannian metric.NEWLINENEWLINESuppose that \(V\) is a smooth vector field on \(X\) which vanishes along a submanifold \(M\). The author defines the notion of the index \(ind_M(V)\) of a vector field \(V\) on \(X\) as \(\int_{Bl(M)} \Phi\) where \([Bl_V(M)]\in H_{n-1}(STX|_M)\) is the element referred to as th \textit{blow-up of \(M\) along \(V\)}. It is shown that \(ind_M(V)\) is an integer which is independent of the metric.NEWLINENEWLINENew proofs of the aforementioned formula of Chern for \(\chi(M)\) are given.