$S^{2}$   and  $\mathbb{R}^{2}$ satisfies the [Poincare  Bendixon theorem](https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Bendixson_theorem)  but this theorem is not satisfied by higher dimensional spheres  or  Euclidean spaces.

For  a  related MSE post see the following.


http://math.stackexchange.com/questions/861231/extension-of-poincar%C3%A9-bendixson-theorem-to-mathbbr3