TITLE: Is there a Poincare-Hopf Index theorem for non compact manifolds?
QUESTION [16 upvotes]: Does Poincare-Hopf index theorem generalizes in any way to non compact manifolds ? In particular, I am interested in the case of a smooth vector field on a cylinder $\mathbb{T}_1\times\mathbb{R}$? If so, are there some additional assumption that one has to impose on a vector field  considered (maybe it should vanish outside some compact set or decay very fast "at infinity"?). Sorry if the question is silly - I know the Hodge index theorem only from very elementary sources (Arnold's book on ODEs and Wikipedia). 
Motivation (physical digression): A friend of mine tries to model action of a cardiac tissue in a heart and its soundings. One of his aims is to understand phenomena called "spiral waves" (they are believed to be partially responsible for hearth attacks). I don't know the details but  those "spiral waves" can be described by some ODE defined on a domain which is closely related to the real geometry of considered tissue. From the information about indexes of singular points of a corresponding vector field it is possible to deduce some qualitative information about occurrence of this phenomenon.

REPLY [13 votes]: Suppose $M$ has empty boundary. Let $U\subset M$ be an open set with compact closure whose topological boundary contains no zero of the continuous vector field $X$ on $M$. Suppose  $X$ is smooth and hence generates a local semiflow $f_t$, $t \geq 0$.
For sufficiently small $t>0$ the map $f_t$$\colon U \to M$ is defined and has a "fixed point index"  $I(f_t, U)$  (see A. Dold, ``Lectures on Algebraic Topology,'' Die
  Grundlehren der matematischen Wissenschaften Bd. 52.  Springer, New
  York (1972)). 
 It can be shown that the integer  $i(X,U):=I(f_t, U)$ is independent of $t$ and $U$, and is stable under perturbation of $X$. 
If $X$ is not smooth,  approximate it by a sequence of smooth fields $X_j$ and define  $I(X,U):= \lim_{j\to \infty} I(X_j,U)$.  
If  $X$ has only finitely many zeros in $U$ and none on $U\cap \partial $, then $I(X, U)$ is the sum of their Poincare-Hopf indices.
If $M$ has nonempty boundary, this work if at every boundary point $p$ there is an integral curve  $u\colon [0,\epsilon)\to M$ with initial point $u(0)=p$.