From the ratio of the SFF to the FFF, we obtain in a given direction. The maximum and minimum normal curvatures at a point are the principal curvatures (\kappa_1, \kappa_2). Their product (K = \kappa_1 \kappa_2) is the Gaussian curvature , and their average (H = (\kappa_1 + \kappa_2)/2) is the mean curvature . 4. The Theorema Egregium and Intrinsic Geometry The most profound moment in any classical differential geometry lecture is Gauss’s Theorema Egregium (Remarkable Theorem): Gaussian curvature depends only on the First Fundamental Form and its derivatives . In other words, (K) is an intrinsic invariant. A being living on a surface can determine (K) by measuring lengths and angles alone, without ever looking into the surrounding 3D space.
with (L = \mathbfx uu \cdot \mathbfN), (M = \mathbfx uv \cdot \mathbfN), (N = \mathbfx_vv \cdot \mathbfN), where (\mathbfN) is the unit normal. The SFF measures how the surface deviates from its tangent plane. lectures on classical differential geometry pdf
[ II = L, du^2 + 2M, du, dv + N, dv^2, ] From the ratio of the SFF to the
[ I = E, du^2 + 2F, du, dv + G, dv^2, ]
[ \int_S K , dA = 2\pi \chi(S), ]
where (E = \mathbfx_u \cdot \mathbfx_u), (F = \mathbfx_u \cdot \mathbfx_v), (G = \mathbfx_v \cdot \mathbfx_v). The FFF is the Riemannian metric induced by the ambient Euclidean space. It allows us to compute arc lengths of curves on the surface, angles between tangent vectors, and areas—all without leaving the surface. Two surfaces with the same FFF are said to be ; they are intrinsically identical, even if shaped differently in space (e.g., a plane and a rolled-up sheet of paper). 3. Measuring Bending: The Second Fundamental Form and Curvatures The FFF tells us about the surface’s metric, but not how it bends in space. For that, we introduce the Second Fundamental Form (SFF): A being living on a surface can determine