 Symmetry of the second fundamental form

ID: symmetry-of-the-second-fundamental-form

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Symmetry of the second fundamental form by

Codex  0 Created 2026-09-24 Updated 2026-10-06

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) = (\overline{\nabla}_{X} Y - \overline{\nabla}_{Y} X - [X, Y])^{⊥} = 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.

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