[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/second-fundamental-form-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/second-fundamental-form-wikipedia\/","headline":"Second fundamental form – Wikipedia","name":"Second fundamental form – Wikipedia","description":"Quadratic form related to curvatures of surfaces In differential geometry, the second fundamental form (or shape tensor) is a quadratic","datePublished":"2015-03-03","dateModified":"2015-03-03","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bcc5b91d573a599cc0b6426ff90ea80310f39b1e","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bcc5b91d573a599cc0b6426ff90ea80310f39b1e","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/second-fundamental-form-wikipedia\/","about":["Wiki"],"wordCount":6598,"articleBody":"Quadratic form related to curvatures of surfacesIn differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by II{displaystyle mathrm {I!I} } (read “two”). Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold.Table of ContentsSurface in R3[edit]Motivation[edit]Classical notation[edit]Physicist’s notation[edit]Hypersurface in a Riemannian manifold[edit]Generalization to arbitrary codimension[edit]See also[edit]References[edit]External links[edit]Surface in R3[edit]  Definition of second fundamental formMotivation[edit]The second fundamental form of a parametric surface S in R3 was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface at the origin. Then f and its partial derivatives with respect to x and y vanish at (0,0). Therefore, the Taylor expansion of f at (0,0) starts with quadratic terms:z=Lx22+Mxy+Ny22+higher order terms,{displaystyle z=L{frac {x^{2}}{2}}+Mxy+N{frac {y^{2}}{2}}+{text{higher order terms}},,}and the second fundamental form at the origin in the coordinates (x,y) is the quadratic formLdx2+2Mdxdy+Ndy2.{displaystyle L,dx^{2}+2M,dx,dy+N,dy^{2},.}For a smooth point P on S, one can choose the coordinate system so that the plane z = 0 is tangent to S at P, and define the second fundamental form in the same way.Classical notation[edit]The second fundamental form of a general parametric surface is defined as follows. Let r = r(u,v) be a regular parametrization of a surface in R3, where r is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of r with respect to u and v by ru and rv. Regularity of the parametrization means that ru and rv are linearly independent for any (u,v) in the domain of r, and hence span the tangent plane to S at each point. Equivalently, the cross product ru \u00d7 rv is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:n=ru\u00d7rv|ru\u00d7rv|.{displaystyle mathbf {n} ={frac {mathbf {r} _{u}times mathbf {r} _{v}}{|mathbf {r} _{u}times mathbf {r} _{v}|}},.}The second fundamental form is usually written asII=Ldu2+2Mdudv+Ndv2,{displaystyle mathrm {I!I} =L,du^{2}+2M,du,dv+N,dv^{2},,}its matrix in the basis {ru, rv} of the tangent plane is[LMMN].{displaystyle {begin{bmatrix}L&M\\M&Nend{bmatrix}},.}The coefficients L, M, N at a given point in the parametric uv-plane are given by the projections of the second partial derivatives of r at that point onto the normal line to S and can be computed with the aid of the dot product as follows:L=ruu\u22c5n,M=ruv\u22c5n,N=rvv\u22c5n.{displaystyle L=mathbf {r} _{uu}cdot mathbf {n} ,,quad M=mathbf {r} _{uv}cdot mathbf {n} ,,quad N=mathbf {r} _{vv}cdot mathbf {n} ,.}For a signed distance field of Hessian H, the second fundamental form coefficients can be computed as follows:L=\u2212ru\u22c5H\u22c5ru,M=\u2212ru\u22c5H\u22c5rv,N=\u2212rv\u22c5H\u22c5rv.{displaystyle L=-mathbf {r} _{u}cdot mathbf {H} cdot mathbf {r} _{u},,quad M=-mathbf {r} _{u}cdot mathbf {H} cdot mathbf {r} _{v},,quad N=-mathbf {r} _{v}cdot mathbf {H} cdot mathbf {r} _{v},.}Physicist’s notation[edit]The second fundamental form of a general parametric surface S is defined as follows.Let r = r(u1,u2) be a regular parametrization of a surface in R3, where r is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of r with respect to u\u03b1 by r\u03b1, \u03b1 = 1, 2. Regularity of the parametrization means that r1 and r2 are linearly independent for any (u1,u2) in the domain of r, and hence span the tangent plane to S at each point. Equivalently, the cross product r1 \u00d7 r2 is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:n=r1\u00d7r2|r1\u00d7r2|.{displaystyle mathbf {n} ={frac {mathbf {r} _{1}times mathbf {r} _{2}}{|mathbf {r} _{1}times mathbf {r} _{2}|}},.}The second fundamental form is usually written asII=b\u03b1\u03b2du\u03b1du\u03b2.{displaystyle mathrm {I!I} =b_{alpha beta },du^{alpha },du^{beta },.}The equation above uses the Einstein summation convention.The coefficients b\u03b1\u03b2 at a given point in the parametric u1u2-plane are given by the projections of the second partial derivatives of r at that point onto the normal line to S and can be computed in terms of the normal vector n as follows:b\u03b1\u03b2=r\u03b1\u03b2\u00a0\u00a0\u03b3n\u03b3.{displaystyle b_{alpha beta }=r_{,alpha beta }^{  ,gamma }n_{gamma },.}Hypersurface in a Riemannian manifold[edit]In Euclidean space, the second fundamental form is given byII(v,w)=\u2212\u27e8d\u03bd(v),w\u27e9\u03bd{displaystyle mathrm {I!I} (v,w)=-langle dnu (v),wrangle nu }where \u03bd{displaystyle nu } is the Gauss map, and d\u03bd{displaystyle dnu } the differential of \u03bd{displaystyle nu } regarded as a vector-valued differential form, and the brackets denote the metric tensor of Euclidean space.More generally, on a Riemannian manifold, the second fundamental form is an equivalent way to describe the shape operator (denoted by S) of a hypersurface,II(v,w)=\u27e8S(v),w\u27e9n=\u2212\u27e8\u2207vn,w\u27e9n=\u27e8n,\u2207vw\u27e9n,{displaystyle mathrm {I} !mathrm {I} (v,w)=langle S(v),wrangle n=-langle nabla _{v}n,wrangle n=langle n,nabla _{v}wrangle n,,}where \u2207vw denotes the covariant derivative of the ambient manifold and n a field of normal vectors on the hypersurface.  (If the affine connection is torsion-free, then the second fundamental form is symmetric.)The sign of the second fundamental form depends on the choice of direction of n (which is called a co-orientation of the hypersurface – for surfaces in Euclidean space, this is equivalently given by a choice of orientation of the surface).Generalization to arbitrary codimension[edit]The second fundamental form can be generalized to arbitrary codimension. In that case it is a quadratic form on the tangent space with values in the normal bundle and it can be defined byII(v,w)=(\u2207vw)\u22a5,{displaystyle mathrm {I!I} (v,w)=(nabla _{v}w)^{bot },,}where (\u2207vw)\u22a5{displaystyle (nabla _{v}w)^{bot }} denotes the orthogonal projection of covariant derivative \u2207vw{displaystyle nabla _{v}w} onto the normal bundle.In Euclidean space, the curvature tensor of a submanifold can be described by the following formula:\u27e8R(u,v)w,z\u27e9=\u27e8II(u,z),II(v,w)\u27e9\u2212\u27e8II(u,w),II(v,z)\u27e9.{displaystyle langle R(u,v)w,zrangle =langle mathrm {I} !mathrm {I} (u,z),mathrm {I} !mathrm {I} (v,w)rangle -langle mathrm {I} !mathrm {I} (u,w),mathrm {I} !mathrm {I} (v,z)rangle .}This is called the Gauss equation, as it may be viewed as a generalization of Gauss’s Theorema Egregium.For general Riemannian manifolds one has to add the curvature of ambient space; if N is a manifold embedded in a Riemannian manifold (M,g) then the curvature tensor RN of N with induced metric can be expressed using the second fundamental form and RM, the curvature tensor of M:\u27e8RN(u,v)w,z\u27e9=\u27e8RM(u,v)w,z\u27e9+\u27e8II(u,z),II(v,w)\u27e9\u2212\u27e8II(u,w),II(v,z)\u27e9.{displaystyle langle R_{N}(u,v)w,zrangle =langle R_{M}(u,v)w,zrangle +langle mathrm {I} !mathrm {I} (u,z),mathrm {I} !mathrm {I} (v,w)rangle -langle mathrm {I} !mathrm {I} (u,w),mathrm {I} !mathrm {I} (v,z)rangle ,.}See also[edit]References[edit]Guggenheimer, Heinrich (1977). “Chapter 10. Surfaces”. Differential Geometry. Dover. ISBN\u00a00-486-63433-7.Kobayashi, Shoshichi & Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 2 (New\u00a0ed.). Wiley-Interscience. ISBN\u00a00-471-15732-5.Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume 3). Publish or Perish. ISBN\u00a00-914098-72-1.External links[edit]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/second-fundamental-form-wikipedia\/#breadcrumbitem","name":"Second fundamental form – Wikipedia"}}]}]