# What happens to the underlying geometry when a lower dimension matrix is embedded in higher dimension?

For example, we can represent a rotation in the xy plane as

$$R=\left\{ R(\theta)=\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}, 0 \leq \theta \leq 2\pi \right\}$$

But this matrix R can be embedded in a rotation in 3D, for instance, to represent the rotation around the z axis as

$$R=\left\{ R(\theta)=\begin{pmatrix} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ \ 0 & 0 & 1 \end{pmatrix}, 0 \leq \theta \leq 2\pi \right\}$$

What happens to the underlying geometry when you embed a smaller matrix into a larger matrix. Specifically, we know that a rotation in 2D is isomorphic to the circle, what happens to this geometry?