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In 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 hypersurface in a Riemannian manifold and a smooth choice of the unit normal vector at each point.

Surface in R³

Definition of second fundamental form

Motivation

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}}+mathrm {scriptstyle {{ }higher{ }order{ }terms}} ,}

and the second fundamental form at the origin in the coordinates x, y is the quadratic form

Ldx2+2Mdxdy+Ndy2.{displaystyle L,{text{d}}x^{2}+2M,{text{d}}x,{text{d}}y+N,{text{d}}y^{2}.,}

For a smooth point P on S, one can choose the coordinate system so that the coordinate z-plane is tangent to S at P and define the second fundamental form in the same way.

Classical notation

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 r_u and r_v. Regularity of the parametrization means that r_u and r_v 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 r_u × r_v is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:

n=ru×rv|ru×rv|.{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 as

II=Ldu2+2Mdudv+Ndv2,{displaystyle mathrm {I!I} =L,{text{d}}u^{2}+2M,{text{d}}u,{text{d}}v+N,{text{d}}v^{2},,}

its matrix in the basis {r_u, r_v} 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⋅n,M=ruv⋅n,N=rvv⋅n.{displaystyle L=mathbf {r} _{uu}cdot mathbf {n} ,quad M=mathbf {r} _{uv}cdot mathbf {n} ,quad N=mathbf {r} _{vv}cdot mathbf {n} .}

Physicist's notation

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^α by r_α, α = 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 × r₂ is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:

n=r1×r2|r1×r2|.{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 as

II=bαβduαduβ.{displaystyle mathrm {I!I} =b_{alpha beta },{text{d}}u^{alpha },{text{d}}u^{beta }.,}

The equation above uses the Einstein Summation Convention.
The coefficients b_αβ at a given point in the parametric (u¹, u²)-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:

bαβ=r,αβ  γnγ.{displaystyle b_{alpha beta }=r_{,alpha beta }^{ gamma }n_{gamma }.}

Hypersurface in a Riemannian manifold

In Euclidean space, the second fundamental form is given by

II(v,w)=−⟨dν(v),w⟩ν{displaystyle mathrm {I!I} (v,w)=-langle dnu (v),wrangle nu }

where ν{displaystyle nu } is the Gauss map, and dν{displaystyle dnu } the differential of ν{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{displaystyle S}) of a hypersurface,

II(v,w)=⟨S(v),w⟩n=−⟨∇vn,w⟩n=⟨n,∇vw⟩n,{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 ∇vw{displaystyle nabla _{v}w} denotes the covariant derivative of the ambient manifold and n{displaystyle 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{displaystyle 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

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 by

II(v,w)=(∇vw)⊥,{displaystyle mathrm {I!I} (v,w)=(nabla _{v}w)^{bot },}

where (∇vw)⊥{displaystyle (nabla _{v}w)^{bot }} denotes the orthogonal projection of covariant derivative ∇vw{displaystyle nabla _{v}w} onto the normal bundle.

In Euclidean space, the curvature tensor of a submanifold can be described by the following formula:

⟨R(u,v)w,z⟩=⟨II(u,z),II(v,w)⟩−⟨II(u,w),II(v,z)⟩.{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{displaystyle N} is a manifold embedded in a Riemannian manifold (M,g{displaystyle M,g}) then the curvature tensor RN{displaystyle R_{N}} of N{displaystyle N} with induced metric can be expressed using the second fundamental form and RM{displaystyle R_{M}}, the curvature tensor of M{displaystyle M}:

⟨RN(u,v)w,z⟩=⟨RM(u,v)w,z⟩+⟨II(u,z),II(v,w)⟩−⟨II(u,w),II(v,z)⟩.{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

• First fundamental form


• Gaussian curvature


• Gauss–Codazzi equations


• Shape operator


• Third fundamental form


• Tautological one-form

References

• 
  Guggenheimer, Heinrich (1977). "Chapter 10. Surfaces". Differential Geometry. Dover. ISBN 0-486-63433-7..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  


• 
  Kobayashi, Shoshichi & Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 2 (New ed.). Wiley-Interscience. ISBN 0-471-15732-5.
  


• 
  Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume 3). Publish or Perish. ISBN 0-914098-72-1.
  

External links

• Steven Verpoort (2008) Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects from Katholieke Universiteit Leuven.