Second fundamental form

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 $\mathrm {I\!I}$ (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:

$z=L{\frac {x^{2}}{2}}+Mxy+N{\frac {y^{2}}{2}}+\mathrm {\scriptstyle {{\ }higher{\ }order{\ }terms}} ,$

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

$L\,{\text{d}}x^{2}+2M\,{\text{d}}x\,{\text{d}}y+N\,{\text{d}}y^{2}.\,$

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:

$\mathbf {n} ={\frac {\mathbf {r} _{u}\times \mathbf {r} _{v}}{|\mathbf {r} _{u}\times \mathbf {r} _{v}|}}.$

The second fundamental form is usually written as

$\mathrm {I\!I} =L\,{\text{d}}u^{2}+2M\,{\text{d}}u\,{\text{d}}v+N\,{\text{d}}v^{2},\,$

its matrix in the basis {r_u, r_v} of the tangent plane is

${\begin{bmatrix}L&M\\M&N\end{bmatrix}}.$

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:

$L=\mathbf {r} _{uu}\cdot \mathbf {n} ,\quad M=\mathbf {r} _{uv}\cdot \mathbf {n} ,\quad N=\mathbf {r} _{vv}\cdot \mathbf {n} .$

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:

$\mathbf {n} ={\frac {\mathbf {r} _{1}\times \mathbf {r} _{2}}{|\mathbf {r} _{1}\times \mathbf {r} _{2}|}}.$

The second fundamental form is usually written as

$\mathrm {I\!I} =b_{\alpha \beta }\,{\text{d}}u^{\alpha }\,{\text{d}}u^{\beta }.\,$

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:

$b_{\alpha \beta }=r_{\alpha \beta }^{\ \ \gamma }n_{\gamma }.$

Hypersurface in a Riemannian manifold

In Euclidean space, the second fundamental form is given by

$\mathrm {I\!I} (v,w)=-\langle d\nu (v),w\rangle \nu$

where $\nu$ is the Gauss map, and $d\nu$ the differential of $\nu$ 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,

$\mathrm {I} \!\mathrm {I} (v,w)=\langle S(v),w\rangle n=-\langle \nabla _{v}n,w\rangle n=\langle n,\nabla _{v}w\rangle n,$

where $\nabla _{v}w$ 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

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

$\mathrm {I\!I} (v,w)=(\nabla _{v}w)^{\bot },$

where $(\nabla _{v}w)^{\bot }$ denotes the orthogonal projection of covariant derivative $\nabla _{v}w$ onto the normal bundle.

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

$\langle R(u,v)w,z\rangle =\langle \mathrm {I} \!\mathrm {I} (u,z),\mathrm {I} \!\mathrm {I} (v,w)\rangle -\langle \mathrm {I} \!\mathrm {I} (u,w),\mathrm {I} \!\mathrm {I} (v,z)\rangle .$

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 $R_{N}$ of $N$ with induced metric can be expressed using the second fundamental form and $R_{M}$, the curvature tensor of $M$:

$\langle R_{N}(u,v)w,z\rangle =\langle R_{M}(u,v)w,z\rangle +\langle \mathrm {I} \!\mathrm {I} (u,z),\mathrm {I} \!\mathrm {I} (v,w)\rangle -\langle \mathrm {I} \!\mathrm {I} (u,w),\mathrm {I} \!\mathrm {I} (v,z)\rangle .$

See also

• First fundamental form
• Gaussian curvature
• Gauss–Codazzi equations
• Shape operator

References

• Guggenheimer, Heinrich (1977). "Chapter 10. Surfaces". Differential Geometry. Dover. ISBN 0-486-63433-7.
• Kobayashi, Shoshichi; Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 2 (New ed.). Wiley-Interscience. ISBN 0-471-15732-5. {{cite book}}: Unknown parameter |lastauthoramp= ignored (|name-list-style= suggested) (help)
• 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.