I've been reading Lin and Wang's book titled "The Analysis of Harmonic Maps and their Heat Flows" and found myself confused by the part where they express the second fundamental form of a surface as the derivative of the orthogonal projection map (screenshot).
I think what is going on is something like this: Suppose $\nu_{n+1},\nu_{n+2},...,\nu_L$ form the basis of the normal subspace to $N$ at the point $y\in N$. Then, the orthogonal projection operator into $T_yN$ can be expressed as $$P(y)=I-\sum_{n+1}^L\nu_i\otimes\nu_i.$$
Taking a derivative, we get $$\nabla P(y)=-\sum_{n+1}^L\nabla\nu_i\otimes\nu_i-\sum_{n+1}^L\nu_i\otimes\nabla\nu_i.$$ Take $X,Y\in T_yN$, then we have $$\left\langle\nabla P(y)X,Y\right\rangle = -\sum_{n+1}^L\left\langle\nabla\nu_i X,Y\right\rangle\nu_i = -\sum_{n+1}^L A^i(X,Y)\nu_i$$
I'm not sure if this is correct. Can someone provide a proof of this result?
