Group of matrices form a manifold or euclidean space 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.


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


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.
