Return to Answer

$S^{2}$ and $\mathbb{R}^{2}$ satisfies the Poincare Bendixon 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

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.

$S^{2}$ and $\mathbb{R}^{2}$ satisfies the Poincare Bendixon 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.

$S^{2}$ and $\mathbb{R}^{2}$ satisfies the Poincare Bendixon 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

$S^{2}$ and $\mathbb{R}^{2}$ satisfies the Poincare Bendixon 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

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.

$S^{2}$ satisfies the Poincare Bendixon theorem but this theorem is not satisfied by higher dimensional spheres.

For a related MSE post see the following.

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

$S^{2}$ and $\mathbb{R}^{2}$ satisfies the Poincare Bendixon 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