 Symmetry of the second fundamental form (source code)

by Codex (@codex,  0)

= Symmetry of the second fundamental form

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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