# 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 immersed submanifold in a Riemannian manifold.

## Surface in R$^3$
### Motivation

The second fundamental form of a parametric surface S in R$^3$ 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:
    $z=L{\frac {x^{2}}{2}}+Mxy+N{\frac {y^{2}}{2}}+{\text{higher order terms}}\,,$
and the second fundamental form at the origin in the coordinates (x,y) is the quadratic form
    $L\,dx^{2}+2M\,dx\,dy+N\,dy^{2}\,.$
For a smooth point P on S, one can choose the coordinate system so that the plane z = 0 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 R$^3$, 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
    $\mathrm {I\!I} =L\,du^{2}+2M\,du\,dv+N\,dv^{2}\,,$
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:
    $L=\mathbf {r} _{uu}\cdot \mathbf {n} \,,\quad M=\mathbf {r} _{uv}\cdot \mathbf {n} \,,\quad N=\mathbf {r} _{vv}\cdot \mathbf {n} \,.$
For a signed distance field of Hessian H, the second fundamental form coefficients can be computed as follows:
    $L=-\mathbf {r} _{u}\cdot \mathbf {H} \cdot \mathbf {r} _{u}\,,\quad M=-\mathbf {r} _{u}\cdot \mathbf {H} \cdot \mathbf {r} _{v}\,,\quad N=-\mathbf {r} _{v}\cdot \mathbf {H} \cdot \mathbf {r} _{v}\,.$

### Physicist's notation

The second fundamental form of a general parametric surface S is defined as follows.
Let r = r(u$^1$,u$^2$) be a regular parametrization of a surface in R$^3$, 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 r$_1$ and r$_2$ are linearly independent for any (u$^1$,u$^2$) in the domain of r, and hence span the tangent plane to S at each point. Equivalently, the cross product r$_1$ × r$_2$ 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
    $\mathrm {I\!I} =b_{\alpha \beta }\,du^{\alpha }\,du^{\beta }\,.$
The equation above uses the Einstein summation convention.
The coefficients b$_{αβ}$ at a given point in the parametric u$^1$u$^2$-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,
    $\mathrm {I} \!\mathrm {I} (v,w)=\langle S(v),w\rangle =-\langle \nabla _{v}n,w\rangle =\langle n,\nabla _{v}w\rangle \,,$
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
    $\mathrm {I\!I} (v,w)=(\nabla _{v}w)^{\bot }\,,$
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:
    $\langle R(u,v)w,z\rangle =\mathrm {I} \!\mathrm {I} (u,z)\mathrm {I} \!\mathrm {I} (v,w)-\mathrm {I} \!\mathrm {I} (u,w)\mathrm {I} \!\mathrm {I} (v,z).$
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:
    $\langle R_{N}(u,v)w,z\rangle =\langle R_{M}(u,v)w,z\rangle +\langle \mathrm {I} \!\mathrm {I} (u,z),\mathrm {I} \!\mathrm {I} (v,w)\rangle -\langle \mathrm {I} \!\mathrm {I} (u,w),\mathrm {I} \!\mathrm {I} (v,z)\rangle \,.$