aka 2-dimensional manifold

Regular surfaces

– Often one refers to regular surfaces when saying surface (depends on context).

Regular surface: Mathematical definition of a regular surface as a subset from R^3 where every point lies on an open subset with a Coordinate chart

The graph of a real-valued function is a regular surface: proof

Examples:

• Sphere

Useful theorem to proof things are regular surfaces: Implicit function theorem

Fundamental theorem of surfaces (Uniqueness result): A smooth surface is determined up to Rigid motion by its first and second fundamental forms.

General 2d Manifolds

like Projective plane..

They have been classified up to homeomorphism by Moebious into:

• Orientable (which are n-tori, where n is the Genus)
• Non-orientable which are connected-sum'd copies of the real Projective plane

Whitney embedding theorem

Functions defined on a surface

Deffinition of a differentiable function with domain (defined on) a surface. Definition for functions with values (range) on a surface. He then gives some examples of differentiable functions. It is enough to check differentiability around a point using just a single chart, no need to check in all possible charts (as is the definition of differentiability), as one can prove that the former implies the later.

Tangent space