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 form
$$
A(X,Y)=(\overline\nabla_XY)^\perp.
$$
It is symmetric because $\overline\nabla$ is torsion-free and the Lie bracket of tangent vector fields remains tangent:
$$
A(X,Y)-A(Y,X)
=\bigl(\overline\nabla_XY-\overline\nabla_YX\bigr)^\perp
=[X,Y]^\perp=0.
$$
This is the <symmetry of the second fundamental form>.

Solution

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

The second fundamental form is symmetric because the ambient Euclidean connection is torsion-free:
$$
A(X,Y)-A(Y,X)=\bigl(\overline\nabla_XY-\overline\nabla_YX-[X,Y]\bigr)^\perp=0.
$$


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