Lecture 10 - First Fundamental Form

#math

Definition

Let $S \subset R^{3}$ be a regular surface. Let $p$ be a point on the surface $S$ . We say $T_{p} S$ is the set of all tangent vectors at $p$ of $S$ . $T_{p} S$ is called the tangent space of $S$ at $p$ and is the vector subspace of $R^{3}$ spanned by the vectors $X_{u}, X_{v}$ .

A tangent vector at $p \in S$ can be obtained from curves $γ (t)$ passing through $p$ . If $σ$ is your mapping from $R^{2} \to S$ then your tangent vector $T$ has the representation:

T_{p} = {(σ^{- 1} \circ γ)^{'} |}_{p}

That is if $γ (t) = X (u (t), v (t))$ , where $X : R^{2} \to S$ then $\frac{\partial γ}{\partial t} = X_{u} u_{t} + X_{v} v_{t}$ is a tangent vector of $p$ (if evaluated at $p$ ). The following example should help clarify this notation:

Example

Let $S$ be the round unit sphere embedded in $R^{3}$ the surface we are considering. Then $γ (t) = (c o s (t), s i n (t), 0)$ is a curve on the sphere that passes through $(1, 0, 0)$ , but this representation is not unique and depends on the chart we assign to the sphere. One could instead consider the same curve but instead work on a spherical coordinate system and then the curve $γ (t)$ would have the representation $γ (t) = (1, \frac{π}{2}, t)$ . Thus our tangent vector's representation depends on the coordinate system.
The tangent vector associated with $γ (t)$ under the regular cartesian coordinate system has representation:

- s i n (t) \frac{\partial}{\partial x} + c o s (t) \frac{\partial}{\partial y} + 0 \frac{\partial}{\partial z}

While under the spherical coordinate system is:

0 \frac{\partial}{\partial r} + 1 \frac{\partial}{\partial θ} + 0 \frac{\partial}{\partial φ}

This is not all that surprising. These are indeed the same tangent vectors. To check a special property of vectors is that we can apply on functions to get a scalar value. Lets take $f (p) = x^{2} + y^{2} + z^{2}$ which is equivalent to $f (p) = r^{2}$ and let us evaluate this on $p = (1, 0, 0)$ (which is $t = 0$ for both the cartesian and spherical representation of the curve). Then the tangent vector $T_{(1, 0, 0)}$ under the cartesian coordinate system applied to $f$ is:

\begin{aligned} {(- s i n (t) \frac{\partial}{\partial x} + c o s (t) \frac{\partial}{\partial y} + 0 \frac{\partial}{\partial z}) (x^{2} + y^{2} + z^{2}) |}_{(1, 0, 0)} & = {(- 2 x s i n (t) + 2 c o s (t) y) |}_{(1, 0, 0)} \\ = (- 2 s i n (0)) \\ = 0. \end{aligned}

While under the spherical coordinate system $p = (1, \frac{π}{2}, 0)$ thus we get:

\begin{aligned} {(0 \frac{\partial}{\partial r} + 1 \frac{\partial}{\partial θ} + 0 \frac{\partial}{\partial φ}) (r^{2}) |}_{p = (1, \frac{π}{2}, 0)} & = 0. \end{aligned}

So thus indeed the action of the tangent vector on the scalar field is the same despite different representations!

Do note that we cannot compare tangent vectors on different points of the manifold. That is if you have a tangent vector $a \in T_{p}$ and another vector $b \in T_{q}$ where $p \neq q$ then you cannot say anything about the relationship of $a$ and $b$ .

Note: (Write about what the basis of the tangent space here)

Definition

The normal vector at $p$ is defined to be:

N_{X} = \frac{X_{u} \times X_{v}}{| | X_{u} \times X_{v} | |}

Orientable Surface

Definition

Let $S$ be a regular surface in $R^{3}$ . Let it be covered by a collection of coordinate patches $X_{i} : X_{i} : U_{i} \to R^{3}$ . Now we have that:

N_{i} = \frac{\partial_{u} X_{i} \times \partial_{v} X_{i}}{| | \partial_{u} X_{i} \times \partial_{v} X_{i} | |}

We say $S$ is orientable if $\forall q \in U_{i} \cap U_{j}$ we have $N_{i} (q) = N_{j} (q)$ .

Theorem

$S$ is orientable if the transition maps have a positive Jacobian.

First Fundamental Form

Definition

Let $S$ be a regular surface, $p \in S$ . The first fundamental form, $I_{p} (\cdot, \cdot)$ is the restriction of the usual dot product in $R^{3}$ to the tangent plane, i.e:

I_{p} (a, b) = a \cdot b =< a, b >

Let $a, b \in T_{p} S$ with the following representation:

\begin{aligned} a & = a_{1} X_{u} + a_{2} X_{v} \\ b & = b_{1} X_{u} + a_{2} X_{v} \end{aligned}

then:

\begin{aligned} {< a, b > |}_{p} & = {(a_{1} b_{1} X_{u} \cdot X_{u} + a_{1} b_{2} X_{u} \cdot X_{v} + a_{2} b_{1} X_{v} \cdot X_{u} + b_{1} b_{2} X_{u} \cdot X_{v}) |}_{p} \end{aligned}

Now let $g_{11} = X_{u} \cdot X_{u}, g_{12} = g_{21} = X_{u} \cdot X_{v}, g_{22} = X_{v} \cdot X_{v}$ then we can represent our first fundamental form as a symmetric matrix:

I_{p} (a, b) = (\begin{matrix} a_{1} & a_{2} \end{matrix}) (\begin{matrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{matrix}) (\begin{matrix} b_{1} \\ b_{2} \end{matrix})

This is know as the quadratic form.

Let $d u, d v$ be the dual vectors of $X_{u}, X_{v}$ .

Now suppose $a = λ X_{u} + μ X_{v}$ , then we can represent the first fundamental form in the following way:

\begin{aligned} < a, a > & = λ^{2} X_{u} \cdot X_{u} + 2 λ μ X_{u} \cdot X_{v} + μ^{2} \cdot X_{v} \cdot X_{v} \\ = E d u^{2} + 2 F d u d v + G d v^{2} |_{a} \end{aligned}

We write the first fundamental form in the following manner:

d s^{2} + E d u^{2} + 2 F d u d v + G d v^{2}

This is know as the line element.
Arc length formula over the interval $[t_{0}, t_{1}]$ can be expressed as:

\begin{aligned} S (t) & = \int_{t_{0}}^{t_{1}} \sqrt{g_{11} u^{'} (τ)^{2} + 2 g_{12} u^{'} (τ) v^{'} (τ) + g_{22} v^{'} (τ)^{2}} d τ \\ = \int_{t_{0}}^{t_{1}} \sqrt{I_{γ (τ)} (γ^{'} (τ), γ^{'} (τ))} d τ \end{aligned}

Suppose $\tilde{X} (α, β)$ is a reparameterization of $X (u, v)$ with first fundamental form given by:

\tilde{E} d α^{2} + 2 \tilde{F} d α d β + \tilde{G} d β^{2}

Then $X (u, v) = \tilde{X} (α (u, v), β (u, v))$ we can compute $\tilde{E}, \tilde{F}, \tilde{G}$ from $E, F, G$ . (homework)

Definition

Let $S_{1}$ and $S_{2}$ be two surfaces, and $f : S_{1} \to S_{2}$ be a smooth map. $f$ is called a local isometry if it takes any curve in $S_{1}$ to a curve of the same length in $S_{2}$ .

Theorem

A smooth map is a local isometry iff the symmetric bilinear form $< \cdot >_{p}$ and $f^{*} < \cdot >_{p}$ are equal for all $p \in S_{1}$ .

Definition

If $S_{1}$ and $S_{2}$ are surfaces, a conformal map $f : S_{1} \to S_{2}$ is a local diffeomorphism such that $γ_{1}$ , $γ_{2}$ are two curves on $S_{1}$ that intersect at $p \in S_{1}$ if $\tilde{γ_{1},} \tilde{γ_{2}}$ are their images under $f$ , the angle of intersection of $γ_{1}, γ_{2}$ at p is equal to the angle of intersection of $\tilde{γ_{1}}, \tilde{γ_{2}}$ at $f (p)$ .