Lecture 11 - Second Fundamental Form

#math

Theorem

A local diffeomorphism $f : S_{1} \to S_{2}$ is conformal iff there exists a function:

λ : S_{1} \to R

such that $f^{*} < v, w >_{p} = λ (p) < v, w >_{p}$
If $λ (p) = 1$ we have a local isometry.

Corollary

A local diffeomorphism $f : S_{1} \to S_{2}$ is conformal iff for any surface patch $X$ of $S_{1}$ the first fundamental form $X$ of $S_{1}$ and of the patch $\tilde{X}$ of $S_{2}$ are proportional.

Definition

The area $A_{X} (R)$ of part of $X (R)$ of a surface patch:

X : U \to R^{3}

corresponding to a region $R \subset U$ is:

A_{X (R)} = \int_{R} | | X_{u} \times X_{v} | | d u d v

Proposition

If $X_{u} \cdot X_{u} = E, X_{u} \cdot X_{v} = F, X_{v} \cdot X_{v} = G$ then:

| | X_{u} \times X_{v} | | = \sqrt{E G - F^{2}}

Proof

$| | X_{u} \times X_{v} | |^{2} = (X_{u} \times X_{v} \cdot X_{u} \times X_{v}) = | | X_{u} | |^{2} | | X_{v} | |^{2} s i n^{2} θ = E G s i n^{2} θ$
where $θ$ is the angle between $X_{u}$ and $X_{v}$
It is known that $c o s θ = \frac{X_{u} \cdot X_{v}}{| | X_{u} | | | | X_{v} | |} = \frac{F}{\sqrt{E G}}$
This completes our proof.

Proposition

The area of surface patch is unchanged/invariant under reparameterization.

Proof

Let $X : U \to R^{3}$ be a surface patch and let $\tilde{X} : \tilde{U} \to R^{3}$ be a reparameterization of $X$ . Let $ϕ : \tilde{U} \to U$ be the reparameterization map:

\tilde{X} (\tilde{u}, \tilde{v}) = X \circ ϕ (u, v)

Note that:

{\tilde{X}}_{\tilde{u}} \times {\tilde{X}}_{\tilde{v}} = d e t (D ϕ) X_{u} \times X_{v}

Thus:

\begin{aligned} \int_{\tilde{R}} | | {\tilde{X}}_{\tilde{u}} \times {\tilde{X}}_{\tilde{v}} | | d \tilde{u} d \tilde{v} & = \int_{\tilde{R}} | d e t (D ϕ) | | | X_{u} \times X_{v} | | d \tilde{u} d \tilde{v} \\ = \int_{R} | | X_{u} \times X_{v} | | d u d v \end{aligned}

Second Fundamental Form

Let $X$ be a surface patch for an oriented surface with standard unit normal $\hat{n}$ . Consider $(u, v) \to (u + Δ u, v + Δ v)$ , then the surface moves away from the tangent plane through $X (u, v)$ by a distance of:

(X (u + Δ u, v + Δ v) - X (u, v)) \cdot \hat{n}

By Taylor expansion we obtain that:

\begin{aligned} (X (u + Δ u, v + Δ v) - X (u, v)) \cdot \hat{n} & = (X_{u} Δ u + X_{v} Δ v + \frac{1}{2} (X_{u u} (Δ u)^{2} + 2 X_{u v} Δ u Δ v + X_{v v} (Δ v)^{2} + . . .)) \cdot \hat{n} \\ = \frac{1}{2} [X_{u u} \cdot \hat{n} (Δ u)^{2} + 2 (X_{u v} \cdot \hat{n}) Δ u Δ v + X_{v v} \cdot \hat{n} (Δ v)^{2}] \end{aligned}

Now if we let $X_{u u} \cdot \hat{n} = L, X_{u v} \cdot \hat{n} = M, X_{v v} \cdot \hat{n} = N$ then we obtain our second fundamental form:

Definition

Let $S_{1}$ be a surface and $X : U \to R^{3}$ be a parameterization of $S_{1}$ , then the second fundamental form is:

\frac{1}{2} (L d u^{2} + 2 M d u d v + N d v^{2})

where $X_{u u} \cdot \hat{n} = L, X_{u v} \cdot \hat{n} = M, X_{v v} \cdot \hat{n} = N$ .