2 Let M be a surface in Ra and p e M Let K p be the Gauss cu

2. Let M be a surface in Ra and p e M. Let K (p) be the Gauss curvature of M at p E M. Prove that, for any linearly indepedent vectors v, w E Tp(M), we have where Sp is the Shape-operator

Solution

Assume that v and w are linearly independent vectors.

Then we can write

S_p(v) = av + bw

and S_p(w) = cv + dw.so that

matrix a c

b d

is the matrix of S with respect to v and w.

we have, the Gauss curvature K(p) = det(S(p)) {Result}

consider

S_p(v)XS_p(w) =( av +b w) X (cv+dw)

= det(a c

b d) (vXw)

=K(p) (vXw) [by result K(p)=det(S_p)]

Hence S_p(v)XS_p(w) = K(p) (vXw).