Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 59 / 1 / b / Solution

Use metric compatibility and differentiate the projector:Projecting the derivative and lower tensor index kills the first term, by part (a). The remaining term is the extrinsic curvature:The second projection may be inserted in the definition of $K$, since differentiating $n^{\beta }{n}_{\beta }=-1$ gives $n^{\beta }{\nabla }_{\alpha }{n}_{\beta }=0$.