im working on an seminar on minimal surfaces.Now i got a defintion of a regular surface, while the Professor of the book i'm reading defined it in a totally different way. In my next paragraph i want to introduce the covariant derivative, but the defintion for the covariant derivative he gives is dependent on his notion of surface. See the problem for yourself:
First my definition of a regular surface:
Definiton (regular surface) A subset $S$ of $\mathbb R^3$ is called a regular surface, if for every point $p \in S$ there exists an open set $V \in \mathbb R^3$, an open set $U \subset \mathbb R^2$ and a function $X:S \rightarrow V\cap S$, so that the following holds true:
- $X(u,v)=\begin{matrix} x(u,v)\\ y(u,v)\\ z(u,v) \end{matrix}$
$x,y,z \in C^\infty$
- $X:U\longrightarrow V\cap S$ is an homeomorphism.
- For all points $p \in S$ the differential $dX_{q}:\mathbb{R^{\mbox{2}}}\longrightarrow\mathbb{R^{\mbox{3}}}$ is an injective function.
The book im reading, which is was refering to earlier, is called differential geometry written by Wolfgang Kühnel. He's not using regular surfaces the way i do, instead he defines the following:
Definiton (piece of a regular surface) Let $U \in \mathbb R^2$ be an open set. A piece of a parameterized regular surface is an immersion $f:U \rightarrow \mathbb R^3$, $(u,v) \to f(u,v)$. $f$ is it's parametrisation.
A piece of an unparameterized regular surface is the equivalence class of pieces of a parameterized regular surfaces, while $f$, $\tilde{f}$ are equivalent, if there exsists a diffeomorphism $\phi$ with $f \circ \phi = f$.
Now to my question: Professor Künhel defines the covariant derivate in the terms of his defintion of surface above:
Definition(covariant derivative) Let $f:U \rightarrow \mathbb R^{n+1}$ be piece of a regular surface, let $X:U \rightarrow \mathbb R^{n+1},Y:U \rightarrow R^{n+1} $ be tangential vectorfields onto f. Then we call $\nabla_X Y :=(D_x Y)^{tang.}= D_X Y- <D_x Y, \nu> \nu $, where $\nu$ is the Gauss map and $D_X Y$ is the directional derivative of $Y$ along $X$.
I did define the surface in terms of charts and open sets, while he did it in "one" piece. How do i translate the definition of the covariant derivative in terms of my definiton of a surface? Or would it be wiser to introduce his defintion of a ** piece of a surface** and explain the covariant derivative in that terms. Then somehow explain what it means in terms of my regular surface?
