# Why is the shape operator self adjoint regarding the first fundamental form?

Why is the shape operator $A: \mathbb R^2 \to \mathbb R^2$self adjoint regarding the first fundamental form? I see it is written as a fact everywhere but cannot find a proof for it.

• See Wolfgang Kuhnel's book on Differential Geometry (third edition), page 67. – Isomorphism Mar 7 '18 at 10:43