Shape Tensor

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 $\mathrm$ (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.

Surface in R³

Motivation

The second fundamental form of a parametric surface in was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, , and that the plane is tangent to the surface at the origin. Then and its partial derivatives with respect to and vanish at (0,0). Therefore, the Taylor expansion of ''f'' at (0,0) starts with quadratic terms:

: $z=L\frac + Mxy + N\frac + \text\,,$

and the second fundamental form at the origin in the coordinates is the quadratic form

: $L \, dx^2 + 2M \, dx \, dy + N \, dy^2 \,.$

For a smooth point on , one can choose the coordinate system so that the plane is tangent to at , 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 be a regular parametrization of a surface in , where is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of with respect to and by and . Regularity of the parametrization means that and are linearly independent for any in the domain of , and hence span the tangent plane to at each point. Equivalently, the cross product is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors :

:$\mathbf = \frac \,.$

The second fundamental form is usually written as

:$\mathrm = L\, du^2 + 2M\, du\, dv + N\, dv^2 \,,$

its matrix in the basis of the tangent plane is

:$\begin L&M\\ M&N \end \,.$

The coefficients at a given point in the parametric -plane are given by the projections of the second partial derivatives of at that point onto the normal line to and can be computed with the aid of the dot product as follows:

:$L = \mathbf_ \cdot \mathbf\,, \quad M = \mathbf_ \cdot \mathbf\,, \quad N = \mathbf_ \cdot \mathbf\,.$

For a signed distance field of Hessian , the second fundamental form coefficients can be computed as follows:

:$L = -\mathbf_u \cdot \mathbf \cdot \mathbf_u\,, \quad M = -\mathbf_u \cdot \mathbf \cdot \mathbf_v\,, \quad N = -\mathbf_v \cdot \mathbf \cdot \mathbf_v\,.$

Physicist's notation

The second fundamental form of a general parametric surface is defined as follows.

Let be a regular parametrization of a surface in , where is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of with respect to by , . Regularity of the parametrization means that and are linearly independent for any in the domain of , and hence span the tangent plane to at each point. Equivalently, the cross product is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors :

:$\mathbf = \frac\,.$

The second fundamental form is usually written as

:$\mathrm = b_ \, du^ \, du^ \,.$

The equation above uses the Einstein summation convention.

The coefficients at a given point in the parametric -plane are given by the projections of the second partial derivatives of at that point onto the normal line to and can be computed in terms of the normal vector as follows:

:$b_ = r_^ n_\,.$

Hypersurface in a Riemannian manifold

In Euclidean space, the second fundamental form is given by

:$\mathrm(v,w) = -\langle d\nu(v),w\rangle\nu$

where is the Gauss map, and the differential of 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 ) 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 denotes the covariant derivative of the ambient manifold and 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 (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
:$\mathrm(v,w)=(\nabla_v w)^\bot\,,$
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 is a manifold embedded in a Riemannian manifold then the curvature tensor of with induced metric can be expressed using the second fundamental form and , the curvature tensor of :
:$\langle R_N(u,v)w,z\rangle = \langle R_M(u,v)w,z\rangle+\langle \mathrm I\!\mathrm I(u,z),\mathrm I\!\mathrm I(v,w)\rangle-\langle \mathrm I\!\mathrm I(u,w),\mathrm I\!\mathrm I(v,z)\rangle\,.$

See also

*First fundamental form
*Gaussian curvature
*Gauss–Codazzi equations
*Shape operator
*Third fundamental form
* Tautological one-form

References

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External links

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

{{curvature

Differential geometry
Differential geometry of surfaces
Riemannian geometry
Curvature (mathematics)
Tensors