On the unit sphere $S^n\subseteq\mathbb R^{n+1}$ choose the outward unit normal $N(x)=x$. For tangent vector fields $X,Y$,
$$
0=X\langle Y,N\rangle
=\langle\overline\nabla_XY,N\rangle+\langle Y,X\rangle,
$$
so
$$
A(X,Y)=-\langle X,Y\rangle N.
$$
If $X,Y$ are orthonormal, the <Gauss equation> gives
$$
\langle R(X,Y)Y,X\rangle
=\langle A(X,X),A(Y,Y)\rangle-|A(X,Y)|^2=1.
$$
Thus the round unit sphere has <sectional curvature> one. Tracing over an orthonormal basis gives its <scalar curvature>
$$
\operatorname{Scal}_{S^n}=n(n-1).
$$

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-115/1/c/solution