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 {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.

Definition of second fundamental form

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 itspartial derivatives with respect to x and yvanish at (0,0). Therefore, the Taylor expansion of f at (0,0) starts with quadratic terms:

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

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.

The second fundamental form of a general parametric surface is defined as follows. Letr = 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 rwith respect to u and v by r_u and r_v. Regularity of the parametrization means that r_u andr_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:

The second fundamental form is usually written as

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

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 toS and can be computed with the aid of the dot product as follows:

The second fundamental form of a general parametric surface S is defined as follows: Letr=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 rwith respect to u^α by r_α, α = 1, 2. Regularity of the parametrization means that r₁ andr₂ 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:

The second fundamental form is usually written as

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:

In Euclidean space, the second fundamental form is given by

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

where {displaystyle nabla _{v}w} denotes the covariant derivative of the ambient manifold and {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 {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).

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

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

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

This is called the Gauss equation, as it may be viewed as a generalization of Gauss'sTheorema Egregium.

For general Riemannian manifolds one has to add the curvature of ambient space; if {displaystyle N} is a manifold embedded in a Riemannian manifold ({displaystyle M,g}) then the curvature tensor {displaystyle R_{N}} of {displaystyle N}with induced metric can be expressed using the second fundamental form and {displaystyle R_{M}}, the curvature tensor of {displaystyle M}: