%!TEX root = 162AB Lecture Notes v2.tex


\chapter{Geometry of Surfaces in $\R^3$}


Recall a \emph{Euclidean frame field} or a \emph{frame field}, constitutes vector fields $\{E_1, E_2, E_3\}$ such that 
\[
E_i\cdot E_j=\delta_{ij}.
\]
Let $\{\theta_1,\theta_2,\theta_3\}$ be the dual frame, that is
\[
\theta_i(E_j)=\delta_{ij}.
\]
The connection $1$-forms $\omega_{ij}$ are defined by 
\[
\nabla_v E_i=\omega_{ij}(v) E_j.
\]

We then have
\[
\begin{split}
&
d\theta_i=\omega_{ij}\wedge\theta_j\\
& d\omega_{ij}=\omega_{ik}\wedge\omega_{kj}.
\end{split}
\]


\section{The fundamental equations}
\begin{definition}{}{}
A \emph{adapted frame field} $\{E_1,E_2,E_3\}$ on a region $\mathscr O$ in $M\subset \R^3$ is a Euclidean frame
field such that $E_3$ is always normal to $M$ (hence $E_1,E_2$ are tangent to $M$).
\end{definition}

\begin{lemma}{}{}
There is an adapted frame field on a region $\mathscr D$ in $M\subset\R^3$ if and only if $\mathscr O$ is orientable  and there exists a non vanishing tangent vector  field on $\mathscr O$.
\end{lemma}
{Example: Cylinder}
Cylinder $M: x^2+y^2=r^2$.

{Example: Sphere}
Sphere  $M: x^2+y^2+z^2=r^2$.


\begin{corollary}{}{} Let $\{E_1,E_2,E_3\}$ be an adapted frame. Then
\[
S(v)=\omega_{13}(v)E_1+\omega_{23} (v)E_2.
\]
\end{corollary}



\begin{theorem}{}{}
Let $\{E_1,E_2,E_3\}$ be an adapted frame field. Then its dual forms and connection forms satisfy
\begin{align*}
&\left\{
\begin{array}{ll}
d\theta_1=\omega_{12}\wedge \theta_2\\
d\theta_2=\omega_{21}\wedge \theta_1
\end{array}\right.& \text{First structural equations}\\[5pt]
&
\omega_{31}\wedge\theta_1+\omega_{32}\wedge\theta_2=0& \text{Symmetry equation}\\[5pt]
& d\omega_{12}=\omega_{13}\wedge\omega_{32} & \text{Gauss equation}\\[5pt]
&\left\{
\begin{array}{ll}
d\omega_{13}=\omega_{12}\wedge \omega_{23}\\
d\omega_{23}=\omega_{21}\wedge \omega_{13}
\end{array}\right.& \text{Codazzi equations}
\end{align*}
\end{theorem}
\begin{note}This is difficult and important.\end{note}
{\S 6.2 Form computations}
   \begin{lemma}
   Let $\phi$ be a $1$-form and let $\mu$ be a $2$-form. Then
   \[
   \begin{split}
   &\phi=\phi(E_1)\theta_1+\phi(E_2)\theta_2\\
   &\mu=\mu(E_1,E_2)\theta_1\wedge\theta_2
   \end{split}
   \]
   \end{lemma}

   \begin{lemma}{}{}
   \[
   \begin{split}
&   \omega_{13}\wedge\omega_{23}=K\theta_1\wedge\theta_2\\
 &\omega_{13}\wedge\theta_2+\theta_1\wedge\omega_{23}=2H\theta_1\wedge\theta_2
 \end{split}
 \]  
   \end{lemma}
   
{The birth of Differential Geometry: \emph{Theorema Egregium!}}
   \begin{corollary}{}{}
   \[
   d\omega_{12}=-K\,\theta_1\wedge\theta_2.
   \]
   \end{corollary}


\begin{definition}{}{}
A \emph{principal frame field} on $M\subset \R^3$ is an adapted frame field $\{E_1,E_2,E_3\}$ such  that at each point $E_1,E_2$ are the principle vectors of $M$.
\end{definition}
\begin{lemma}
If ${\bf p}$ is a non-umbilic  of $M\subset \R^3$, then there exists a principal frame field on some neighborhood of ${\rm p}$ in $M$.
\end{lemma}

If $\{E_1,E_2,E_3\}$ is a principal frame field on $M$, then we have
\[
\begin{split}
\omega_{13}(E_1)=\lambda_1,\qquad &\omega_{13}(E_2)=0\\
\omega_{23}(E_1)=0,\qquad & \omega_{23}(E_2)=\lambda_2
\end{split}
\]
We then have
\begin{theorem}{}{}
\[
\begin{split}
E_1(\lambda_2)=(k_1-k_2)\omega_{12}(E_2)\\
E_2(\lambda_1)=(k_1-k_2)\omega_{12}(E_1)
\end{split}
\]
\end{theorem}



\begin{theorem}
If the shape operator is identically zero, then $M$ is part of a plane in $\R^3$.
\end{theorem}

\begin{lemma}{}{}
If $M$ is all-umbilic surface in $\R^3$, then $M$ has constant Gaussian curvature $K\geq 0$.
\end{lemma}

\begin{theorem}{}{}
If $M\subset \R^3$ is all-umbilic and $K>0$, then $M$ is part of a sphere in $\R^3$ of radius $1/\sqrt K$.
\end{theorem}

\begin{corollary}{}{}
A compact all-umbilic surface $M$ in $\R^3$ is an entire sphere.
\end{corollary}

\begin{theorem}
On every compact surface $M$ in $\R^3$ there is a point which the Gaussian curvature $K$ is strictly positive.
\end{theorem}

\begin{lemma}{}{} ${\bf m}$ be a point of $M\subset \R^3$ such that 
\begin{enumerate}
\item $\lambda_1$ is a local maximum at ${\bf m}$;
\item $\lambda_2$ is a local minimum at ${\bf m}$;
\item $\lambda_1>\lambda_2$.
\end{enumerate}
Then $K\leq 0$. 
\end{lemma}

\begin{theorem}{}{}
If $M$ is a compact surface in $\R^3$ with constant Gaussian curvature $K$, then $M$ is a sphere of radius $1/\sqrt K$.
\end{theorem}


Consider a surface whose first fundamental form is given by
\[
\begin{bmatrix} E&F\\F&G\end{bmatrix}
=
\begin{bmatrix} \frac{1}{(1+\frac K4(x^{2}+y^{2}))^{2}}&0\\[10pt]0& \frac{1}{(1+\frac K4(x^{2}+y^{2}))^{2}}\end{bmatrix}
\]

























\section{Isometries and local isometries}

\begin{definition}{}{}[surface as a metric space] If $p,q$ are points of $M\subset \R^3$, consider the collection of all curve segments $\alpha$ in $M$ from $p$ to $q$. The \emph{intrinsic distance} $\rho(p,q)$ from $p$ to $q$ in $M$ is the greatest lower bound of the lengths $L(\alpha)$ of these curve segments.
\end{definition}


\begin{definition}{}{}
An \emph{ isometry}  $F:M\to\overline M$ of surfaces in $\R^{3}$ is a one-to-one  mapping of $M$ onto $\overline M$ that preserves the dot products of tangent vectors. That is, 
\[
F_*(v)\cdot F_*(W)=v\cdot w.
\]
\end{definition}

\begin{definition}{}{}
A \emph{local isometry} is ...... oh, local.
\end{definition}

\begin{theorem}{}{}
Isometries preserve intrinsic distance.
\end{theorem}

\section{Intrinsic geometry of surfaces in}
\begin{lemma}{}{}
The connection form $\omega_{12}=-\omega_{21}$ in the only 1-form that satisfies the first structural  equations
\[
d\theta_1=\omega_{12}\wedge \theta_2,\qquad d\theta_2=\omega_{2}\wedge\theta_1.
\]
\end{lemma}

\section{Orthogonal Coordinates}

\begin{definition}{}{}
Frame is called \emph{orthogonal}, if in the first fundamental form, $F=0$.
\end{definition}
\begin{example}
For the unit sphere, we have
\[
E=r^2\cos^2 v,\qquad F=0,\qquad G=r^2.
\]
\end{example}



\[
K=\frac{-1}{\sqrt{EG}}\left\{\left(\frac{(\sqrt G)_u}{\sqrt E}\right)_u+\left(\frac{(\sqrt E)_v}{\sqrt G}\right)_v\right\}.
\]






\section{Integration and Orientation}






We have already talked about the integration over a differential form in \S 4.6. In this section, we shall extend the definition \emph{and} we shall introduce the \emph{area form}, which is important and is related to the orientation. 

\vspace{.1in}

\begin{note}Let 
\[
\begin{bmatrix}
E&F\\F&G
\end{bmatrix}
\]
be the first fundamental form. Define 
\[
\sqrt{EG-F^2} du\wedge dv
\]
to be the area form. 
\end{note}
{Orientability}
Recall that a surface is orientable, if and only if there is a non-zero 2-form on the surface.


\begin{theorem}{}{}
$M$ is orientable if and only if the area form is well-defined.
\end{theorem}

{Example 1: area of sphere}
\[
E=r^2\cos^2 v, \qquad F=0,\qquad G=r^2.
\]

{Example 2: area of torus}
\[
E=r^2,\qquad F=0,\qquad G=(R+r\cos u)^2.
\]


{Example 3: Bugle surface}
\[
E=1,\qquad F=0,\qquad G=h^2=c^2e^{-2u/c}.
\]



\begin{definition}{}{}
A \emph{paving} of a region $\mathscr P$ in a surface $M$ consists of a finite member of patchlike 2-segments ${\bf x}_1,\cdots {\bf x}_k$ whose images fill $M$ in such a way that each point of $M$ is in at most one set ${\bf x}_i(\mathscr R^0_i)$.
\end{definition}

\begin{definition}{}{}
Let $\nu$ be a 2-form   on  pavable oriented region $\mathscr P$ in a surface. The integral  of $\nu$ over $\mathscr P$ is
\[
\iint_{\mathscr P}\nu=\sum_i\iint_{{\bf x}_i} \nu.
\]
\end{definition}

\section{Total Curvature}


\begin{definition}
The total curvature is defined by
\[
\int_M K\, dM.
\]
\end{definition}
{Example: constant curvature}
If the Gauss curvature is a constant, then
\[
\int_M K dM=K\int_M dM=K{\rm Area}(M).
\]
For example, the total curvature of a sphere of radius $r$ is given by 
\[
\frac{1}{r^2} \, 4\pi r^2=4\pi.
\]

The total curvature of the bugle surface is
\[
-\frac{1}{c^2}\cdot 2\pi c^2=-2\pi.
\]
{Example: torus $T$}
For the torus, we know that the Gauss curvature is given by
\[
K=\frac{\cos u}{r(R+r\cos u)}.
\]

The volume form is given by
\[
dM=r(R+r\cos u) dudv.
\]

Therefore
\[
\int_T KdT=\int^\pi_{-\pi}\int_{-\pi}^\pi\cos u \,dudv=0
\]
{Example: Catenoid}
The Gauss curvature is
\[
K=\frac{-1}{c^2\cosh^4(u/c)},
\]
and
\[ 
dM=\sqrt{EG}=c\cosh^2(u/c).
\]
Thus the total
\[
\int_{M(a)} K dM=-\int^a_{-a}\int_0^{2\pi}\frac{1}{c\cosh^2 (u/c)} dudv=-4\pi\tanh(a/c)
\]
As $a\to\infty$, we have
\[
\int_MKdM=-4\pi.
\]

\begin{definition}{}{}
Let $M$ and $N$ be surfaces oriented by area forms $dM$ and $dN$. The \emph{Jacobian} of a mapping $F: M\to N$ is the real-valued function $J_F$ such that 
\[
F^*(dN)=J_F dM.
\]
\end{definition}


\begin{theorem}The Gauss curvature $K$ of an oriented surface $M\subset \R^3$ is the Jacobian of its Gauss map.
\end{theorem}

\begin{definition}{}{}
Let $v,w$ be unit tangent vectors at a point of an oriented surface $M$. A number $\phi$ is an oriented angle from $v$ to $w$ provided 
\[
w=\cos\phi v+\sin\phi J(v),
\]
where $J(v)=U\times v$.
\end{definition}


\begin{lemma}{}{}
Let $\alpha: I\to M$ be a curve in an oriented surface $M$. If $V$ and $W$ are nonvanishing tangent vector fields on $\alpha$, there is a differentiable function $\phi$ on $I$ such that  for each $t$ in $I$, $\phi(t)$ is an oriented angle  from $V(t)$ to $W(t)$.
\end{lemma}


\begin{theorem}{}{}
A simple closed curve $\alpha$ in $\R^3$ has total curvature 
\[
\int_\alpha \kappa ds\geq 2\pi.
\]
\end{theorem}


\section{Congruence of surfaces}
\begin{theorem}{}{}
If ${\bf F}$ is a Euclidean isometry such that ${\bf F}(M)=\overline M$, then the restriction of ${\bf F}$ to $M$ is an isometry of surfaces. Moreover, if $M$ and $\overline M$ are suitably oriented, then $F$ preserves shape operators, that is,
\[
F_*(S(v))=\overline S(F_* (v)).
\]
\end{theorem}


\begin{theorem}{}{}
Let $M$ and $\overline M$ be oriented surfaces in $\R^3$. Let $F: M\to\overline M$ be an isometry of oriented surfaces in $\R^3$ that preserves shape operators, so
\[
F_*(S(v))=\overline S(F_* (v))
\]
for any tangent vectors to $M$. Then $M$ and $\overline M$ are congruent; in fact, there is a Euclidean isometry ${\bf F}$ such that ${\bf F}|_M=F$.
\end{theorem}
