$S^{2}$   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.


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

As another example: For $n>8$ there is  no a $n$-dimensional subvector  space of $M_{n}(\mathbb{R})$ which all non zero elements are invertible matrix. The only possible $n$ are $n=1,2,4,8$. Such subvector spaces correspond to matrix representation of real numbers, complex numbers, Quaternions  and  Cayley numbers, respectively.