# Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem:

THEOREM. Let $M$ be a compact, connected, two-dimensional manifold of class $C^2$. Let $\alpha: \mathbb R \times M \to M$ be a $C^2$ action of the reals on $M$. Let $\Omega \subset M$ be an $\alpha$-minimal set. Then $\Omega$ must be one of the following:
$a)$ a singleton consisting of a fixed point
$b)$ a single, closed orbit homeomorphic to $S^1$
$c)$ all of $M$ which is homeomorphic to a torus $T^2$

For the proof, the author considers three cases for $\Omega$:
$1)$ $\Omega$ is a single fixed point
$2)$ $\Omega$ is a closed orbit which is homeomorphic to $S^1$, i.e. a periodic orbit
$3)$ $\Omega$ is a set which contains neither fixed points nor closed orbits.
The cases $1)$ and $2)$ are trivial and for the case $3)$, the author considers two cases:
$3)'$ $\Omega$ has non-empty interior
$3)''$ $\Omega$ has empty interior and, being closed, is nowhere dense.
For the case $3)'$ the author claims that:

In this case, since the set of interior points is invariant and $\Omega$ is minimal, the set of boundary points must be empty. Thus $\Omega$ is open and closed and must be all of $M$. It follows from a result of Kneser $[4, p. 153]$ that since $M$ contains neither fixed points nor closed orbits it must be homeomorphic to $T^2$.

I have a big problem with this part of proof since the reference that the author mentioned, is a paper written by H. Kneser by the title "Regulare Kurvenscharen auf den Ringfiachen" published in "Mathematische Annalen, vol. 91 (1924), pp. 135-154" which is written in German.
I am trying to find an article or a book that contains the results of this paper in English language.

• Does the author allow $M$ to have boundary? – user98602 Jul 24 '16 at 18:06
• @MikeMiller: Yes. It is possible. – hamid kamali Jul 24 '16 at 18:41

Consider the vector field $X$ that generates the flow. Because the flow must necessarily preserve the boundary components, if there are boundary components, they must be circular orbits. So there are none. If $X$ has no zeroes, then $\chi(M) = 0$; this follows either from a definition of $\chi(M)$ that involves counting degrees of zeroes of vector fields (Poincare-Hopf theorem) or from noting that $\chi(M)$ is the Lefschetz number of any map homotopic to the identity, and your flow (for small time) is homotopic to the identity and has no fixed points.