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Shape Operator Symmetry

  1. May 23, 2006 #1
    :confused: Please, help me to solve that task related to Shape Operator.

    We have surface [tex]S[/tex] and its normal [tex]N[/tex]. Alse we have surface patch [tex]r : U -> S[/tex] in local coordinates [tex]r_1, r_2, ..., r_n[/tex]. Shape operator (Weingarten Linear Operator) is defined as follow:
    [tex]L_p : T_{r(p)}S -> T_{r(p)}S[/tex], where [tex]T_{r(p)}S[/tex] - tangent plane to surface.
    It is known that [tex]L_p(w) = -D_vN(p), w \in T_{r(p)}S[/tex].
    It is necessary to proof, that shape operator is symmetrical.

    There is theorem that shows that shape operator is symmetrical [tex]L_p : T_pS -> T_pS, L_p(v)*w = v*L_p(w)[/tex], but on the surface. For patch we need to prof that matrix is symmetrical or something like that..

    Can anyone lead me to right direction?
    Thank you.
     
  2. jcsd
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