Trisha Shetty

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 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.

Contents

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 x 2 2 + M x y + N y 2 2 +   h i g h e r   o r d e r   t e r m s ,

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

L d x 2 + 2 M d x d y + N 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 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:

n = r u × r v | r u × r v | .

The second fundamental form is usually written as

I I = L d u 2 + 2 M d u d v + N d v 2 ,

its matrix in the basis {ru, rv} of the tangent plane is

[ L M M N ] .

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 = r u u n , M = r u v n , N = r v v 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 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:

n = r 1 × r 2 | r 1 × r 2 | .

The second fundamental form is usually written as

I I = b α β d u α d u β .

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 α β = r α β     γ n γ .

Hypersurface in a Riemannian manifold

In Euclidean space, the second fundamental form is given by

I I ( v , w ) = d ν ( v ) , w ν

where ν is the Gauss map, and d ν the differential of ν 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,

I I ( v , w ) = S ( v ) , w n = v n , w n = n , v w n ,

where 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

I I ( v , w ) = ( v w ) ,

where ( v w ) denotes the orthogonal projection of covariant derivative 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 = I I ( u , z ) , I I ( v , w ) I I ( u , w ) , I I ( v , z ) .

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 :

R N ( u , v ) w , z = R M ( u , v ) w , z + I I ( u , z ) , I I ( v , w ) I I ( u , w ) , I I ( v , z ) .

References

Second fundamental form Wikipedia


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