I'm reading an article and it describes signals on a sphere. I dont understand how SO(3) isn't considered a signal on the sphere. And why is the sphere referred to as S2, when spheres are 3D manifolds? the statement is below:
"whereas the space of moves for the plane (2D translations) is itself isomorphic to the plane, the space of moves for the sphere (3D rotations) is a different, three-dimensional manifold called SO(3). It follows that the result of a spherical correlation (the output feature map) is to be considered a signal on SO(3), not a signal on the sphere, S2."
$S^2$ refers only to the surface of the (ordinary) sphere; it may typically be embedded in three-dimensional space, but that doesn't make it a three-dimensional manifold; it's two-dimensional. If you take a small section of it, it looks like a section of a plane, not a spatial object. After all, you can embed a circle (also known as $S^1$) in three-dimensional space too, but that doesn't make it a three-dimensional manifold either. It is one-dimensional.
As regards the other question: The space of translations is equivalent to the plane itself, because any point in the plane can be identified with the translation that moves the origin to that point, or, alternatively, with the translation that moves that point to the origin.
However, the space of rotations is not equivalent to (the surface of) the sphere $S^2$. Any rotation can be characterized by an axis around which the rotation is counter-clockwise—which can be identified with a point on the sphere—and the amount of the rotation. There are thus three degrees of freedom, whereas there are only two degrees of freedom on the sphere (say, longitude and latitude).
In addition, there is a wrinkle, in that the above characterization is a "double cover"; any rotation of $\theta$ around an axis $v$ can likewise be characterized as a rotation of $2\pi-\theta$ around the axis $-v$. With that duplication accounted for, the space of rotations is often called $\operatorname{SO}(3)$, which stands for the special orthogonal group of $3$-by-$3$ matrices. (Unsurprisingly, $\operatorname{SO}(2)$ represents the rotations of the circle.)