= Tangential derivative of a normal field
{title2=$\langle\nabla_X\eta,Y\rangle=-\langle\eta,II(X,Y)\rangle$}
For the ambient <Levi-Civita connection>, a normal field $\eta$, and tangent fields $X,Y$, differentiate $\langle\eta,Y\rangle=0$ using metric compatibility. The tangential part of $\nabla_XY$ is orthogonal to $\eta$, giving the displayed identity. It controls how an ambient derivative of a normal vector can develop a tangential component, and is the vector-valued version of the <shape operator> relation.
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