Lecture 12 - Gauss Map

#math

The first fundamental form describes the intrinsic geometry, while the second fundamental form describes how the surfaces is embedded in a higher dimension.

First take a surface $X (u, v)$ and we push it inwards a distance along its normal to get a one parameter family of surfaces:

Y (u, v, t) = X (u, v) - t N (u, v)

Thus we obtain:

\begin{aligned} Y_{u} & = X_{u} - t N_{u} \\ Y_{v} & = X_{v} - t N_{v} \end{aligned}

Computing the first fundamental form for this surface $E d u^{2} + 2 F d u d v + G d v^{2}$ , computing the rate of change of the first fundamental form (and multiplying by half) we obtain:

\begin{aligned} \frac{1}{2} \frac{\partial}{\partial t} (E d u^{2} + 2 F d u d v + G d v^{2}) & = - (X_{u} \cdot N_{u}) d u^{2} - (X_{u} \cdot N_{v} + X_{v} \cdot N_{u}) d u d v - (X_{v} \cdot N_{v}) d v^{2} \\ = L d u^{2} + 2 M d u d v + N d v^{2} \end{aligned}

That is we obtain the second fundamental form by looking at how the first fundamental form changes w.r.t a deformation of pushing the surface in the direction of the normal!

Gauss Map

Let $X$ be a surface $X \subset R^{3}$ . When we have $N$ as its normal. Then what we are doing is defining a Gauss map. That is we are defining the map:

N : X \to S^{2}

Where $N$ assigns each point on the surface to a unit vector orthogonal to its surface. We can naturally extend this map on the tangent space of a point $p$ to obtain:

d N_{p} : T_{p} X \to T_{N (p)} S^{2}

Now notice that if we take $< d N_{p} (v), w >$ for $v, w \in T_{p} X$ :

\begin{aligned} < d N_{p} (v), w > & = - < v_{1} N_{u} + v_{2} N_{v}, w_{1} X_{u} + w_{2} X_{v} > \\ = \sum_{i, j = 1, 2} v_{i} w_{j} N_{i} X_{j} \\ = {I I}_{p} (v, w) \end{aligned}

Notice that this is self adjoint, that is: $< d N_{p} (v), w >=< v, d N_{p} (w) >$

We can write ${I I}_{p} (v, v) = v^{T} (\begin{matrix} L_{11} & L_{12} \\ L_{21} & L_{22} \end{matrix}) v$ .

Definition

Let $S$ be a regular orientable surface of $C^{2}$ . A point $p$ on $S$ is called:

Elliptic if $d e t (L_{i j}) > 0$
Hyperbolic if $d e t (L_{i j}) < 0$
Parabolic if $d e t (L_{i j}) = 0$ but not all $L_{i j} = 0$
If $L_{i j} = 0$ for all $i, j$ then its called plane/flat.

Definition

Gaussian curvature is defined by:
$κ = d e t (d N_{p})$

We can write $N_{u} = a_{1}^{1} X_{u} + a_{1}^{2} X_{v}$ and $N_{v} = a_{2}^{1} X_{u} + a_{2}^{2} X_{v}$ thus:

\begin{aligned} d N_{p} (w) & = w_{1} N_{u} + w_{2} N_{v} \\ = w_{1} (a_{1}^{1} X_{u} + a_{1}^{2} X_{v}) + w_{2} (a_{2}^{1} X_{u} + a_{2}^{2} X_{v}) \\ = (a_{1}^{1} w_{1} + a_{2}^{1} w_{2}) X_{u} + (a_{1}^{2} w_{1} + a_{2}^{2} w_{2}) X_{v} \end{aligned}

Thus:

d N_{p} (\begin{matrix} w_{1} \\ w_{2} \end{matrix}) = (\begin{matrix} a_{1}^{1} & a_{2}^{1} \\ a_{1}^{2} & a_{2}^{2} \end{matrix}) (\begin{matrix} w_{1} \\ w_{2} \end{matrix})

Hence:

\begin{aligned} L_{i j} & = N_{i} \cdot X_{j} \\ = (\sum a_{i}^{k} X_{k}) \cdot X_{j} \\ ⟹ L_{i j} & = \sum_{i = 1}^{2} a_{i}^{k} g_{k j} \end{aligned}

where $g_{11} = E, g_{21} = g_{12} = F, g_{22} = G$ .

Thus:

(\begin{matrix} a_{1}^{1} & a_{2}^{1} \\ a_{1}^{2} & a_{2}^{2} \end{matrix}) = {(\begin{matrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{matrix})}^{- 1} (\begin{matrix} L_{11} & L_{12} \\ L_{21} & L_{22} \end{matrix}) = {(\begin{matrix} E & F \\ F & G \end{matrix})}^{- 1} (\begin{matrix} L & M \\ M & N \end{matrix})

$K = k_{1} k_{2} = d e t (a_{j}^{i})$ and $H = \frac{1}{2} T r (D N_{p})$ . Where $H$ is called the mean curvature. Where $k_{1}, k_{2}$ are called the principle curvatures (they are simply the eigenvalues of the $a_{j}^{i}$ matrix.)

Homework

Show the following facts:

$H = \frac{G L - 2 F M + E N}{2 (E G - F^{2})}$ , $K = \frac{L N - M^{2}}{E G - F^{2}}$
Principle curvature are roots of the equation:

k^{2} - 2 H k + K = 0

$k_{1} + k_{2} = 2 H$

Theorem

Let $S$ be a surface with isothermal parameterization (that is $E = G$ ) then $S$ is minimal (that is $H = 0$ ) iff $L = - N$