Overview
One of the first topics of early differential geometry was the description and analysis of surfaces. The fundamental insight about surfaces was that they could be parameterized by two real variables, labeled {% u^1 %} and {% u^2 %} below.
Surface
A surface embedded within three dimensional space is characterized as a set of {% \mathbb{R}^3 %} vectors parameterized by two real variables.
{% \vec{x}(u^1, u^2) = (x_1(u^1, u^2),x_2(u^1, u^2),x_3(u^1, u^2)) %}
The partial derivatives of {% \vec{x} %} are given
{% x_\alpha = \frac{\partial \vec{x}}{u^\alpha} %}
The partial derivatives are tangent vectors to the surface. In this case, there
are two of them.
Example
{% \vec{x}(u^1, u^2) = (u^1 cos (u^2), u^1 sin (u^2), u^1) %}
Curves on the Surface
A curve is a function from {% \mathbb{R} %} to the surface. Given the parameterization of the surface above, this can be written as
{% \vec{x}(t) = \vec{x}(u_1(t), u_2(t)) %}
Metric
Using the notation given above for the partial derivatives of {% \vec{x} %}, the squared distance between two points on the surface can be approximated as
{% ds^2 = (\vec{x}_1 du^1 + \vec{x}_2 du^2) \cdot (\vec{x}_1 du^1 + \vec{x}_2 du^2) %}
Defining a matrix of numbers {% g %} as
{% g_{\alpha \beta} = \vec{x}_\alpha \cdot \vec{x}_\beta %}
We can restate the squared distance as
{% ds^2 = \sum_\alpha \sum _\beta g_{\alpha \beta} du^{\alpha} du ^{\beta} %}
The matrix {% g %} is called the metric on the surface.
Gauss Map and Surface Normal
At every point along a smooth surface, two vectors normal to the surface with unit length can be defined, each pointing in opposite directions. It is typical to assign an orientation to the surface by choosing one of the normals to be the vector defined to be the normal vector to the surface. This assignment of a normal is defined as the gauss map.
{% \mathcal{G} : S \rightarrow \vec{n} %}
The normal can be computed as
{% \frac{X_{u1} \times X_{u2}}{ | X_{u1} \times X_{u2} |} %}
where {% X_{u\alpha} = \frac{\partial X}{\partial u_\alpha} %}