Given a random rotation in n-dimensions, what surface is created?
In 2 dimensions this is obviously the circle. The same is true in 3 dimensions. (Any continuous rotation in 3 dimensions of a random point around the origin will produce a circle or a degenerate point if the rotation axis is through the point).
But in 4-dimensions once could have rotation which is a combination of a rotation in the $xy$ axis combined with a rotation in the $zw$ axis, and if these rotations are not rational ratios of each other, a point a 4D space will never return to its starting point. So this can't be a circle. (What shape is it? A torus? Will it go through all points in space?)
Similarly what shapes are produced in n-dimensions by a continuous n-dimensional rotation of a point around the origin? Is there a general theory behind this?