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

Let $\overline{\nabla }$ be the Euclidean connection and split the ambient tangent bundle along the embedded submanifold as $T{\mathbb{R}}^{n+m}{|}_M=TM\oplus NM$. The second fundamental form is the normal-bundle-valued bilinear formIt is symmetric because $\overline{\nabla }$ is torsion-free and the Lie bracket of tangent vector fields remains tangent:This is the symmetry of the second fundamental form.

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 $A$.

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