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With outward normal $N(x)=x$, the unit sphere has $A(X,Y)=-⟨X,Y⟩N$. The Gauss equation therefore gives sectional curvature one on every tangent two-plane.

With the curvature convention $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$, the Gauss equation for a Euclidean embedded submanifold isThe Codazzi equation iswhere the derivative uses the Levi-Civita connection on tangent arguments and the normal connection on the value of $A$.

Solved by gpt-5.6-sol high.

On the unit sphere $S^n\subseteq {\mathbb{R}}^{n+1}$ choose the outward unit normal $N(x)=x$. For tangent vector fields $X,Y$,soIf $X,Y$ are orthonormal, the Gauss equation givesThus the round unit sphere has sectional curvature one. Tracing over an orthonormal basis gives its scalar curvature

Solved by gpt-5.6-sol high.