= Symmetry of the second fundamental form
The <second fundamental form> of an <embedded submanifold> of a <Riemannian manifold> is symmetric. Its ambient <Levi-Civita connection> is a <torsion-free connection>, so
$$
II(X,Y)-II(Y,X)=\bigl(\overline\nabla_XY-\overline\nabla_YX-[X,Y]\bigr)^\perp=0.
$$
The <Lie bracket of vector fields> tangent to the submanifold is tangent and therefore has zero normal component. This proof applies to a curved ambient manifold as well as Euclidean space.
Back to article page
