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.

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,


 * $$\mathrm I\!\mathrm I(v,w)=\langle S(v),w\rangle n = -\langle \nabla_v n,w\rangle n=\langle n,\nabla_v w\rangle n\,,$$

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
 * $$\mathrm{I\!I}(v,w)=(\nabla_v w)^\bot\,, $$

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.

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 $M$:
 * $$\langle R_N(u,v)w,z\rangle = \langle R_M(u,v)w,z\rangle+\langle \mathrm I\!\mathrm I(u,z),\mathrm I\!\mathrm I(v,w)\rangle-\langle \mathrm I\!\mathrm I(u,w),\mathrm I\!\mathrm I(v,z)\rangle\,.$$