Second fundamental form

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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 \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[edit]

Motivation[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=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[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 × rv 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 {ru, rv} 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[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 uα by rα, 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 bαβ 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[edit]

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[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 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. The eigenvalues of the second fundamental form, represented in an orthonormal basis, are the principal curvatures of the surface. A collection of orthonormal eigenvectors are called the principal directions.

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[edit]

References[edit]

  • Guggenheimer, Heinrich (1977). "Chapter 10. Surfaces". Differential Geometry. Dover. ISBN 0-486-63433-7. 
  • Kobayashi, Shoshichi and 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[edit]