The second fundamental form is a mathematical object used in differential geometry that provides a way to describe how a surface bends in a higher-dimensional space. Specifically, it is associated with a surface \( S \) embedded in a higher-dimensional Euclidean space, such as \(\mathbb{R}^3\).
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For an embedded submanifold of a Riemannian manifold, the vector-valued second fundamental form isHere is the ambient Levi-Civita connection and the superscript denotes projection onto the normal bundle. For a hypersurface in Euclidean space with unit normal , the scalar form is . In a surface parametrization its coefficients are , , and .