Second fundamental form
In differential geometry, the second fundamental form is a symmetric bilinear form defined on the tangent space of a smooth surface embedded in Euclidean space, quantifying the extrinsic curvature by measuring how the surface deviates from its tangent plane in the normal direction.[1] For a parametrized surface given by a position vector \mathbf{r}(u,v), it is expressed in coordinates as II = L\, du^2 + 2M\, du\, dv + N\, dv^2, where the coefficients L = \mathbf{r}_{uu} \cdot \mathbf{n}, M = \mathbf{r}_{uv} \cdot \mathbf{n}, and N = \mathbf{r}_{vv} \cdot \mathbf{n} involve the second partial derivatives of \mathbf{r} dotted with the unit normal vector \mathbf{n}.[2] This form, introduced by Carl Friedrich Gauss in his 1827 work Disquisitiones generales circa superficies curvas, contrasts with the intrinsic first fundamental form, which captures metric properties like lengths and angles on the surface itself.[3][4] The second fundamental form plays a central role in determining key curvature invariants, such as the normal curvature \kappa_n = II / I along a direction, where I is the first fundamental form, providing a signed measure of bending perpendicular to the tangent plane.[2] Its eigenvalues correspond to the principal curvatures, which describe the maximum and minimum curvatures at a point, while the Gaussian curvature K, the product of these principal curvatures, emerges as the determinant of the matrix relating the second and first fundamental forms, linking extrinsic and intrinsic geometry via Gauss's Theorema Egregium.[1] For a surface where the second fundamental form vanishes identically, the surface must be planar, highlighting its role in classifying flat versus curved embeddings.[5] In higher dimensions, the second fundamental form generalizes to hypersurfaces in Riemannian manifolds, serving as the shape operator that encodes how the hypersurface curves relative to the ambient space, with applications in general relativity and geometric analysis.[1] Its symmetry ensures it defines a self-adjoint operator, facilitating spectral decompositions essential for studying surface evolution, minimal surfaces, and variational problems.[1]Surfaces in Euclidean three-space
Historical context and motivation
The study of curved surfaces in three-dimensional space traces its roots to the eighteenth century, with significant precursors in the works of mathematicians like Leonhard Euler and Gaspard Monge. Euler explored the curvature of surfaces through osculating planes and developable surfaces in his 1760 paper "Recherches sur la courbure des surfaces," laying early groundwork for understanding how surfaces deviate from flatness. Monge, often regarded as the father of differential geometry, advanced the analysis of curved surfaces in his 1785 work "Mémoire sur la théorie des déblais et des remblais," where he introduced concepts like lines of curvature and partial differential equations for surface properties, influencing the geometric study of embeddings in Euclidean space.[6] The second fundamental form was formally introduced by Carl Friedrich Gauss in his seminal 1827 paper "Disquisitiones generales circa superficies curvas," presented to the Royal Society of Göttingen. In this work, Gauss developed a quadratic form to quantify the extrinsic geometry of surfaces, building on but distinctly advancing the earlier French school of geometry led by Monge. Gauss's approach emphasized the local embedding of surfaces, distinguishing it from prior efforts by providing a systematic tool for measuring bending independent of coordinate choices.[4][7] Conceptually, the second fundamental form arises from the need to capture how a surface bends away from its tangent plane in the ambient space, contrasting with the first fundamental form, which encodes the intrinsic metric structure measurable solely from within the surface. For instance, a sphere exhibits pronounced extrinsic curvature as points nearby deviate sharply from the tangent plane, while a plane shows none, highlighting the form's role in distinguishing embedded shapes geometrically. This extrinsic perspective motivates its study by revealing how local deviations influence global form, such as in the embedding of abstract manifolds.[8][5] In differential geometry, the second fundamental form serves as a crucial bridge between the local properties of a surface's embedding and its overall shape, enabling the derivation of key invariants like principal curvatures. Its applications extend to physics, particularly in the theory of minimal surfaces where zero mean curvature—derived from this form—describes equilibrium shapes like soap films. In engineering, it underpins shell theory for analyzing thin curved structures under load, informing designs in architecture and aerospace by quantifying bending stresses.[1][9]Definition via parametrization
Consider a smooth oriented surface S in Euclidean three-space \mathbb{R}^3 given by a parametrization \mathbf{r}: U \to \mathbb{R}^3, where U \subset \mathbb{R}^2 is an open set and \mathbf{r}(u,v) is regular, meaning the partial derivatives \mathbf{r}_u = \frac{\partial \mathbf{r}}{\partial u} and \mathbf{r}_v = \frac{\partial \mathbf{r}}{\partial v} satisfy \mathbf{r}_u \times \mathbf{r}_v \neq \mathbf{0}. These partial derivatives form a basis for the tangent plane T_p S at each point p = \mathbf{r}(u,v) \in S. The unit normal vector \mathbf{n} to S at p is then defined as \mathbf{n} = \frac{\mathbf{r}_u \times \mathbf{r}_v}{|\mathbf{r}_u \times \mathbf{r}_v|}, providing an orientation for the surface.[1][10] The second fundamental form II at p is a symmetric bilinear form on the tangent plane T_p S, measuring the extrinsic curvature of S in the normal direction. For tangent vectors X, Y \in T_p S, it is defined as II(X, Y) = -\langle d\mathbf{n}(X), Y \rangle, where d\mathbf{n}: T_p S \to T_{\mathbf{n}(p)} S^2 is the differential of the Gauss map \mathbf{n}: S \to S^2, and \langle \cdot, \cdot \rangle denotes the Euclidean inner product in \mathbb{R}^3. Equivalently, since \mathbb{R}^3 is equipped with its flat Levi-Civita connection \nabla, we have II(X, Y) = \langle \nabla_X Y, \mathbf{n} \rangle, where \nabla_X Y is the directional derivative of the vector field Y (extended suitably off the surface) along a curve \sigma: (-\epsilon, \epsilon) \to S with \sigma(0) = p and \sigma'(0) = X, projected onto the normal \mathbf{n}. This equivalence follows from the fact that the tangential component of \nabla_X Y lies in T_p S and is orthogonal to \mathbf{n}, while the full decomposition \nabla_X Y = (\nabla_X Y)^\top + II(X,Y) \mathbf{n} isolates the normal part.[1][3][10] In the local coordinates (u,v), the second fundamental form is expressed as the quadratic form II = L \, du^2 + 2M \, du \, dv + N \, dv^2, with coefficients \begin{align*} L &= \langle \mathbf{r}{uu}, \mathbf{n} \rangle, \ M &= \langle \mathbf{r}{uv}, \mathbf{n} \rangle, \ N &= \langle \mathbf{r}_{vv}, \mathbf{n} \rangle, \end{align*} where \mathbf{r}_{uu} = \frac{\partial^2 \mathbf{r}}{\partial u^2}, \mathbf{r}_{uv} = \frac{\partial^2 \mathbf{r}}{\partial u \partial v}, and \mathbf{r}_{vv} = \frac{\partial^2 \mathbf{r}}{\partial v^2} are the second partial derivatives of the parametrization. These coefficients, introduced by Gauss in his classical treatment of surface curvature, represent the matrix \begin{pmatrix} L & M \\ M & N \end{pmatrix} of II with respect to the basis \{\mathbf{r}_u, \mathbf{r}_v\}, defining a quadratic form on T_p S.[4][3][10] The second fundamental form is independent of the choice of parametrization in the tensorial sense: under a change of coordinates (u,v) \to (\tilde{u}, \tilde{v}), the coefficients transform via the Jacobian matrix of the coordinate change, ensuring that II behaves as a covariant (0,2)-tensor on S, contravariantly weighted by the inverse of the first fundamental form's metric tensor. This tensorial nature guarantees that geometric quantities derived from II are well-defined on the surface.[1][10]Notations and conventions
The second fundamental form, often denoted as II, can be expressed in classical notation as the quadratic form II = L \, du^2 + 2M \, du \, dv + N \, dv^2, where the coefficients L = \mathbf{r}_{uu} \cdot \mathbf{n}, M = \mathbf{r}_{uv} \cdot \mathbf{n}, and N = \mathbf{r}_{vv} \cdot \mathbf{n} are computed from the second partial derivatives of the position vector \mathbf{r}(u,v) and the unit normal \mathbf{n}.[11] This form, originating from early differential geometry texts, emphasizes the bilinear nature of the form along coordinate directions. In contrast, modern tensor notation represents II as II = h_{ij} \, du^i \, du^j, with components h_{ij} = \langle \partial^2 \mathbf{r} / \partial u^i \partial u^j, \mathbf{n} \rangle for indices i,j = 1,2 corresponding to coordinates u^1 = u, u^2 = v, providing a more general framework for abstract manifolds.[1][12] In physicist's conventions, particularly in general relativity and continuum mechanics, the second fundamental form adopts index notation II = b_{\alpha\beta} \, du^\alpha \, du^\beta, where b_{\alpha\beta} = \partial^2 r^\gamma / \partial u^\alpha \partial u^\beta \, n_\gamma employs Einstein summation over repeated indices \gamma in the embedding space, facilitating computations in curved spacetimes or material deformations.[13] This notation aligns with tensor analysis in physics, where the form describes extrinsic curvature of hypersurfaces, such as spacelike slices in relativity, and surface stresses in mechanics.[13] The choice of normal direction introduces a sign convention for II, as replacing \mathbf{n} with -\mathbf{n} negates all components, thereby flipping the signs of derived mean curvature while leaving Gaussian curvature invariant.[1][12] For closed surfaces, the outward normal typically yields positive mean curvature for convex shapes like spheres, whereas the inward normal reverses this, impacting interpretations in applications such as fluid interfaces or gravitational horizons.[13]| Notation | Form | Coefficients |
|---|---|---|
| Classical | L \, du^2 + 2M \, du \, dv + N \, dv^2 | L = \mathbf{r}_{uu} \cdot \mathbf{n}, etc.[11] |
| Tensor | h_{ij} \, du^i \, du^j | h_{ij} = \langle \partial^2 \mathbf{r} / \partial u^i \partial u^j, \mathbf{n} \rangle[[1]](https://www.cmu.edu/biolphys/deserno/pdf/diff_geom.pdf) |
| Physicist | b_{\alpha\beta} \, du^\alpha \, du^\beta | b_{\alpha\beta} = r^\gamma_{,\alpha\beta} \, n_\gamma[[13]](https://faculty.etsu.edu/gardnerr/5310/5310pdf/dg1-5.pdf) |