To clarify, would an exponential map be a vector within this pi-radius ball? I'm trying to understand the picture the professor sort of showed with his hand but I'm a little lost on it; if it's a pi-radius ball, what does it mean for a shell to be at 2pi? How can we represent tumbling by increasing the vector length, when in doing that, the vector clearly goes outside of the ball?

ShrihanSolo

I believe we can think of it as a rotation about an axis. Every point on the pi-radius sphere has a possible axis, and degree of rotation, i.e. the length of the vector from the point to the center of the sphere (0 to pi). If we extend the ball beyond 2pi, we'll start seeing singularities, e.g., at the 2pi shell.

To clarify, would an exponential map be a vector within this pi-radius ball? I'm trying to understand the picture the professor sort of showed with his hand but I'm a little lost on it; if it's a pi-radius ball, what does it mean for a shell to be at 2pi? How can we represent tumbling by increasing the vector length, when in doing that, the vector clearly goes outside of the ball?

I believe we can think of it as a rotation about an axis. Every point on the pi-radius sphere has a possible axis, and degree of rotation, i.e. the length of the vector from the point to the center of the sphere (0 to pi). If we extend the ball beyond 2pi, we'll start seeing singularities, e.g., at the 2pi shell.