I know you can rotate an arc about an axis of revolution to create a surface of revolution that sits in three dimensional space. Can you take the resulting surface of revolution and rotate it to create a shape that sits in four dimensional space?
For example, let two unit circles overlap and take the inner lens shape, and then create a surface of revolution with the axis of rotation being the y-axis. Then how do you rotate this surface of revolution into four dimensional space?
Another example, is to take a circle, rotate it to create a sphere sitting in three dimensional space and then the question would be how to rotate $S^2$ so that it is higher dimensional and sits in four dimensional space.
