= Gauss equation with reversed curvature convention
{c}
{title2=$\langle R^M(X,Y)Z,W\rangle=\langle R^V(X,Y)Z,W\rangle-\langle II(X,W),II(Y,Z)\rangle+\langle II(Y,W),II(X,Z)\rangle$}
Here $R(X,Y)=D_{[X,Y]}-[D_X,D_Y]$, the negative of the convention in the ordinary <Gauss equation in a curved ambient manifold>. Insert the <Gauss formula> and <Weingarten formula> into this definition and project tangentially to obtain the displayed identity. Positive <sectional curvature> uses $\langle R(X,Y)X,Y\rangle/|X\wedge Y|^2$ with this convention. Thus a Euclidean surface still has <Gaussian curvature> $K=\det A_\xi$; reversing the operator sign without changing the slot order would give the wrong sign.
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