One formulation way to describe a general formula is this:is nonvanishing in a neighborhood of $\partial M$. Suppose that the set of tangencies
Choose an outward normal vector field $X$ to U$along$\partial M$consists of a codimension one submanifold ; now arrange$T$of X$ sothat its direction coincides with $\partial M$U$only in isolated points, so if we can write$\partial M$as the union of two submanifolds X$ to $N$ along $U$, it is a vector field with boundary, isolated singularities. Let $\partial M = B_+ \cup B_-$be the flow sum of the Poincaré-Hopf indices over all singularities where $X$ is orientedoutwardon $B_+$ and inward on $B_-$.. Then the Poincaré-Hopf index for $i(X)$ of $X$ in $M$ is equals the Euler characteristicof $\chi(M) - \chi(B_+)$M$minus$i_+(X)$. However, along the boundary, some of the numbers on$B_+$(which includes$T$) are blown away and lost.Near any zero of$X$there is a mismatch,which can only depend on To regularize the local behavior of situation, modify$X$.From X$ by pushing in the specialcase that negative normal direction.Now $N$ is X$points inward everywhere except in a closed manifold and$M = N$, neighborhood of points wherethat is coincides with the Poincaré-Hopf indexoutward normal. Thus everything cancels out except forlocal contributions given by$i(X)$and$i_+(X)$. 1 Every noncompact manifold admits nonzero vector fields, or more generally, vector fields with any specified set of isolated zeros along with the behavior near that zero. However, if you have information of the behavior of a vector field near infinity, or just in a neighborhood of the boundary of a compact set, there is an index theorem. Perhaps this is the case with your$\mathbb T^! \times \mathbb R$. One formulation is this: let$N^n$be manifold, and let$M, \partial M \subset N$be a compact submanifold. Let$X$be a vector field that is is nonvanishing in a neighborhood of$\partial M$. Suppose that the set of tangencies of$X$to$\partial M$consists of a codimension one submanifold$T$of$\partial M$, so we can write$ \partial M$as the union of two submanifolds with boundary,$\partial M = B_+ \cup B_-$where the flow of$X$is outward on$B_+$and inward on$B_-$. Then the Poincaré-Hopf index for$X$in$M$is$\chi(M) - \chi(B_+)$Here's one proof: triangulate a neighborhood of$N$so that$\partial M$and$T$are subcomplexes, and so that$X$is transverse to the triangulation except near the singularities, in the sense that in any simplex, the foliation defined by$X$is topologically equivalent to the kernel of a linear map in general position of the simplex to$\mathbb R^{n-1}$. Put a$+1$at the barycenter of each triangle of even dimension, and a$-1$at the barycenter of each triangle of odd dimension. Think of$X$as a wind that blows these numbers along, so that after an instant, all numbers (except for exceptions near the zeros of$X$) are inside an$n$-simplex. In any typical simplex, all the signs cancel out. However, the numbers on$B_+$(which includes$T$) are blown away and lost. Near any zero of$X$there is a mismatch, which can only depend on the local behavior of$X$. From the special case that$N$is a closed manifold and$M = N\$, it follows that is the Poincaré-Hopf index.