Second fundamental form

Let $\vec{v}$ and $\vec{w}$ be two tangent vectors. The second fundamental form is:

$$\mathbb{II}(\vec{v}, \vec{w}) = -W_x(\vec{v})\cdot \vec{w}$$

where $W_x$ is the Weingarten map. The second fundamental form is invariant under parameter transformations that preserves the direction of the normal, thus it is independent from a particular surface representation. The latter results from the parameterization invariance of the Weingarten map. When the surface is $C^2$, it is a symmetric bilinear form on the tangent space $\overrightarrow{T_xS}$. Symmetry can be checked using a parameterization:

$$\begin{array}{rcl} \mathbb{II}(\vec{v}, \vec{w}) &=&-W_x(\vec... ...\cdot x_{uv}\\ &=&\mathbb{II}(\vec{w}, \vec{v})\\ \end{array}$$

(where it can be shown that $\vec{n}_u\cdot x_{v} = -\vec{n}\cdot x_{uv}$). In the basis $(x_u, x_v)$, the second fundamental form is entirely defined by the numbers:

$$\begin{array}{rcl} L&=&x_{u}\cdot\vec{n}_u = x_{uu}\cdot\vec{... ...\\ N&=&x_{v}\cdot\vec{n}_v = x_{vv}\cdot\vec{n}\\ \end{array}$$

Thus if $\vec{\tau}\in \overrightarrow{T_{x_0}S}$ has components $(\tau_u, \tau_v)$ in $(x_u, x_v)$, the second fundamental form defines a positive definite quadratic form:

$$\begin{array}{rcl} \mathbb{II}(\vec{\tau},\vec{\tau}) &=& L... ...(\!\!\begin{array}{c} a \\ b \end{array}\!\!\right) \end{array}$$

Where $L$, $M$ and $N$ are the second fundamental coefficients, and are not invariant under a parameter transformation. We will refer to their matrix as the second fundamental matrix. Let $d_{x_0}(u, v)$ be the distance from $\xi(u, v)$ the tangent plane at $x_0 = \xi(u_0, v_0)$:

$$d_{x_0}(u, v) = (\xi(u, v) - x_0)\cdot\vec{n}$$

Taylor's second order expansion of $\xi(u, v)$ is $x_0 + d_ux_u + d_vx_v + \frac{1}{2}(d_u^2x_{uu} + 2d_ud_vx_{uv} + d_v^2x_{vv})$, where $du = u - u_0$ and $dv = v - v_0$. Replacing the surface by its approximation, and because the normal is perpendicular to tangent vectors:

$$\begin{array}{rcl} d_{x_0}(u, v) &=&\frac{1}{2}(d_u^2x_{uu} ... ...{2}\mathbb{II}(d_ux_u + d_vx_v, d_ux_u + d_vx_v)\\ \end{array}$$

Thus $\mathbb{II}(\vec\tau, \vec\tau)$ is the principal part of twice the distance from $\xi(u, v)$ to the tangent plane at $x_0$. The function $\frac{1}{2}\mathbb{II}(\vec\tau, \vec\tau)$ is called the osculating paraboloid at $x_0$. The following is a second order approximation of the surface:

$$\xi(u_0, v_0) + \vec\tau + \frac{1}{2}\mathbb{II}(\vec\tau, \vec\tau)$$

The determinent of the second fundamental matrix determines the nature of the surface in the neighborhood of $(u_0, v_0)$