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Math Help - Torus coordinate patch

  1. #1
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    Question Torus coordinate patch

    Hi,

    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

    PHI(u,v)=((R+rcos)cosv,(R+rcosu)sinv,rsinu)
    Show that the shape operater S on the torus is given by

    S(PHI_u)=PHI_u/r

    S(PHI_v)=(cosu/(R+rcosu))PHI_v

    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?
    Last edited by musicmental85; November 19th 2007 at 01:52 PM. Reason: Got nBAR
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  2. #2
    Super Member Rebesques's Avatar
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    seems to involve some horrific differentiation

    Yeap, that's differential geometry alright.



    If your calculations are OK, you should have

    S(\Phi_u)=-N_u, \ S(\Phi_v)=-N_v.
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