Second Fundamental Form

Differential Geometry · Axiom Academy

LESSON Second Fundamental Form Understanding how surfaces curve in 3D space through the geometry of the Gauss map 1. Definition and Geometric Meaning The second fundamental form is a symmetric bilinear form that measures how the normal vector N changes in response to tangent vectors v and w . Here, S is the shape operator (or Weingarten map), which maps tangent vectors to tangent vectors and encodes the curvature information. The negative sign ensures proper geometric interpretation. 2. Coefficients e, f, g (or L, M, N) Just as the first fundamental form has coefficients E, F, G, the second fundamental form has coefficients denoted either as e, f, g or L, M, N . e (or L): Second derivative in u-direction dotted with normal f (or M): Mixed partial derivative dotted with normal g (or N): Second derivative in v-direction dotted with normal Like the first fundamental form, the second fundamental form can be represented as a symmetric 2×2 matrix with respect to the coordinate basis X u , X v . For any tangent vector v = a·X u + b·X v , the second fundamental form evaluates to: The primary application of the second fundamental form is computing the normal curvature —how much the surface bends in a given direction. Key geometric interpretations: κ n > 0: Surface curves in the same direction as the normal κ n Surface curves opposite to the normal κ n = 0: Direction is asymptotic (no bending)

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