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I have to prove that the Weingarten Map L for the unit sphere is + or - the identity "by computing the L_{ik}in a coordinate patch andraising 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^{-1}Λ= ( -1 0 ) = -Identity

_________( 0 -1 )

The plus identity is obtained similarly by choosing a parametrization with inward pointing normal.

But what does "raising an index" mean?

Thank you.

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# I Computing the Weingarten Map L by raising an index

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