With the curvature convention $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$, the <Gauss equation> for a Euclidean embedded submanifold is
$$
\langle R(X,Y)Z,W\rangle
=\langle A(X,W),A(Y,Z)\rangle
-\langle A(X,Z),A(Y,W)\rangle.
$$
The <Codazzi equation> is
$$
(\nabla_XA)(Y,Z)=(\nabla_YA)(X,Z),
$$
where 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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 115 / 1 / b / Solution

{c}
{wiki=Gauss–Codazzi_equations}

For a Euclidean embedded submanifold, the covariant derivative of the second fundamental form satisfies
$$
(\nabla_XA)(Y,Z)=(\nabla_YA)(X,Z).
$$


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