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

The projection $X\mapsto X_{\parallel }$ is tensorial. Although a covariant derivative is not $C^{\infty }$-linear in its second argument, the extra term in ${\nabla }_{X_{\parallel }}(f{Y}_{\parallel })$ is proportional to $Y_{\parallel }$, whose contraction with the normal vanishes. Thus $K(X,Y)$ is $C^{\infty }$-linear in both arguments and defines a $(0,2)$ tensor on the hypersurface.

Since $n_a{Y}_{\parallel }^a=0$,Therefore the extrinsic curvature is