= Gauss equation in a curved ambient manifold
{c}
{title2=$\langle R^M(v,w)y,x\rangle=\langle R^N(v,w)y,x\rangle+\langle II(v,x),II(w,y)\rangle-\langle II(v,y),II(w,x)\rangle$}
For an <embedded submanifold> of a <Riemannian manifold>, use $R(v,w)y=\nabla_v\nabla_wy-\nabla_w\nabla_vy-\nabla_{[v,w]}y$. The <Gauss formula> and <tangential derivative of a normal field> give
$$
\langle\nabla_v\nabla_wy,x\rangle
=\langle D_vD_wy,x\rangle-\langle II(w,y),II(v,x)\rangle.
$$
Subtract the expression with $v,w$ interchanged and the bracket derivative, whose normal part pairs to zero. This proves the displayed curvature identity. For flat Euclidean ambient space, its ambient curvature term vanishes and one obtains the ordinary <Gauss equation>. Declaring the four-slot convention $R(x,y,v,w)=\langle R(v,w)y,x\rangle$ avoids sign ambiguities when permuting arguments.
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