Group of matrices form a manifold or euclidean space

Question

There is a very interesting question How can a group of matrices form a manifold. From the answers it looks more like group of matrices form euclidean space than a general manifold. I understand that euclidean space is a manifold, but manifold is very general and has curvature. My question is what exactly makes a group of matrices a manifold but not simply a euclidean space.

• I am not a mathematician so please correct me if there is anything wrong with the question or the way I posed it.

Answer 1

Take $SO(2,\mathbb{R})$, for instance. This is the group of the $2\times2$ orthogonal matrices whose determinant is $1$. But then$$SO(2,\mathbb{R})=\left\{\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}\,\middle|\,\theta\in\mathbb R\right\}.$$This can be seen as a circle in $\mathbb{R}^2$. Therefore, it is naturally a manifold, but in no way an Euclidean space.

Answer 2

In the question to which you refer the group of matrices is given a manifold structure by virtue of its embedding in a Euclidean space. So for example you can think of a torus as a two dimensional manifold (with interesting curvature) by using the differentiable structure it gets from the usual embedding as a curved surface in $3$-space. That doesn't make the torus a Euclidean space.