I'm working on a differential geometry question at the moment which seems to involve some horrific differentiation and I'm starting to wonder if I'm approaching it wrong. Just looking for advice?
Cover the torus with the coordinate patch
Show that the shape operater S on the torus is given by
Right after much incorrect calculating I got nBar=(cosucosv,-cosusinv,-sinu)
I need to find the covariant derivative of the vector field ie (vBAR(X_1),vBar(X_2),vBAR(X_3) where vBAR(X_i) are directional derivatives. How do I do this? Where do I find directional derivatives? I thnk I need to choose a curve with a(o)=p and a'(0)=vBAR but how does one choose this?