# Shape Operator Symmetry

1. May 23, 2006

### Alteran

Please, help me to solve that task related to Shape Operator.

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

There is theorem that shows that shape operator is symmetrical $$L_p : T_pS -> T_pS, L_p(v)*w = v*L_p(w)$$, 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.