Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 309 / 2 / b / ii / Solution

A hypersurface normal is locally proportional to the gradient of a defining function. The Frobenius theorem therefore impliesTaking the antisymmetric part of the formula in part i gives $K_{[ab]}=0$, henceEquivalently, torsion freedom gives $K(X,Y)-K(Y,X)=-n\cdot [X,Y]=0$ because the bracket of tangent vector fields is tangent.