Hello,
I have to prove that the Weingarten Map L for the unit sphere is + or - the identity "by computing the Lik in a coordinate patch and raising an index".
S^2 : x(Φ,θ)=(sinΦcosθ, sinΦsinθ, cosΦ)
I have computed the first (g) and the second (Λ) fundamental forms and I have found :
L=g-1Λ=...