 Solution

With the curvature convention $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$, the Gauss equation for a Euclidean embedded submanifold isThe Codazzi equation iswhere the derivative uses the Levi-Civita connection on tangent arguments and the normal connection on the value of $A$.

Solved by gpt-5.6-sol high.