\chapter{Differentiable surface}



We have studied much of the geometry related to curves that analysis permits, now we turn our attention to the geometry of surfaces.


Defining a differentiable curve was pretty straightforward, it was just an differentiable $\mathbb{R}$ function that draws our a path in $\mathbf{R}^n$. We also had the concept of a regular curve to denote differentiable curves that have a non-zero tangent in its Frenet-Serret frame.

Differentiable surfaces are more complicated; we'll define them in their regular form because regular surfaces guarantee a nontrivial tangent plane (similar to a regular curve guaranteeing a nonzero tangent); which will be extremely desirable in our study.

\section{Regular surface}

\begin{definition}[Regular surface]
A \emph{regular surface} is a set $\Sigma \subset \mathbb{R}^3$ such that any $p \in \Sigma$ has a neighborhood $V$ open in $\mathbb{R}^3$ and a differentiable homeomorphism  $\sigma : U \to V \cap \Sigma$ (where $U$ is an open set of $\mathbb{R}^2$) such that for any $q\in U$, its differential $\mathrm{d}\sigma_q : \mathbb{R}^2 \to \mathbb{R}^3$ is injective.
\end{definition}

Why not define regular surfaces by one single differentiable homeomorphism rather than allowing functions for neighborhoods over the surface? Some objects that we intuitively want to consider surfaces require several 'patches of functions' to be constructed, such as a sphere. To make our theory richer, we allow such 'neighborhood patches' to form regular surfaces.

We obviously want these functions to be differentiable because it will allow us to apply vector analysis, and we require them to be homomorphisms so that our coordinate system $U$ acts topologically the same as $V \cap \mathbb{R}^3$

That third condition is what makes the regular surface a regular one, more technically, it ensures that each point always has a nontrivial tangent plane.
How does this condition ensure this? This is easier see when it is restated in the following way.

\begin{proposition}
 $\mathrm{d}\sigma_q : \mathbb{R}^2 \to \mathbb{R}^3$ is injective iff are linearly independent.
\end{proposition}

This condition will be useful when studying tangent planes, specifically, it allows for a standard way to span tangent planes.


We will now prove some theorems that can be used to facilitate the checking that certain types of sets constitute a regular surface. Many of these results require implicit function theorem.


\begin{proposition}
Given a smooth $\mathbb{R}^2$ function defined on an open set, its graph is a regular surface.
	\[ \{ \begin{bmatrix}u \\ v \\ f(u,v) \end{bmatrix}  : \begin{bmatrix}u \\ v\end{bmatrix} \in \mathbb{R}^2 \}   \]
\end{proposition}

By the implicit function theorem, level sets are either regular surfaces or empty sets.

\begin{proposition}
Let $f : U \subset \mathbb{R}^3 \to \mathbb{R}$ be a dfferentiable function and $p \in f(U)$ be a regular (non-critical) point, then $f^{-1}(p)$ is a regular surface.
\end{proposition}

\begin{proposition}
Let $S \subset \mathbb{R}^3$ be a regular surface with $p\in S$, then there exists some neighborhood $V$ of $p$ where $V$ is the graph of either of the differentiable functions $f(x,y),f(x,z),f(y,z)$.
\end{proposition}


\section{Tangent planes}

Imagine considering the derivative on surfaces; we would want this to be irrespective of parametrization and the open sets used.

\begin{proposition}
Let $\Sigma$ be a regular surface containing $p$. Let $\sigma_1$ and $\sigma_2$ parametrize different neighborhoods of $p$ on $\Sigma$, and define $\sigma_1 (U) \cap \sigma_2 (V) = W$. Then $\sigma_{1}^{-1} \circ \sigma_2 : \sigma_{2}^{-1}(W) \to \sigma_{1}^{-1}(W)$ is a diffeomorphism between the coordinate systems.
\end{proposition}



\begin{definition}[Differentiable function on a regular surface]
Let $\Sigma$ be a regular surface. A \emph{differentiable function on $\Sigma$ at $p$} is a function on an open subset of the surface $f : V \subset \Sigma \to \mathbb{R}$, where some parametrization $\sigma$, $\mathbf{f} \circ \sigma$ is differentiable at $\sigma^{-1}(p)$.
\end{definition}
 Due to the previous theorem, the parametrization doesn't matter for differentiability; parametrizations overlapping some neighborhood are diffeomorphic.




\begin{definition}[Tangent plane]
\end{definition}

In differential topology (more abstract, topology-centered study of manifolds than differential geometry) we define tangent spaces, and tangent planes are tangent spaces of points on a surface.





\begin{definition}[Parametrized surface]
	Let $\Sigma$ be a regular surface. A \emph{differentiable function on $\Sigma$ at $p$} is a function $f : V \subset S \to \mathbb{R}$, where some parametrization $\mathbf{x}$, $\mathbf{f} \circ \mathbf{x}$ is differentiable.
A parametrized surface is the image of a differentiable function $f : U \subseteq \mathbb{R}^2 \to \mathbb{R}^3$ whose differential of any point is injective.
\end{definition}


Any point of a regular parametrized surfaces has a neighborhood that is a regular surface.