Solution
= Solution
A hypersurface normal is locally proportional to the gradient of a defining function. The <Frobenius theorem> therefore implies
$$
h_a{}^ch_b{}^d\nabla_{[c}n_{d]}=0.
$$
Taking the antisymmetric part of the formula in part i gives $K_{[ab]}=0$, hence
$$
\boxed{K_{ab}=K_{ba}}.
$$
Equivalently, torsion freedom gives $K(X,Y)-K(Y,X)=-n\mathbin\cdot[X,Y]=0$ because the bracket of tangent vector fields is tangent.