= Solution
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.
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