Surfaces and Submanifolds
The classical theory of surfaces considers the subsets of 3 dimensional euclidean space that are either defined parametrically by a map from a 2 dimensional variety into the 3 dimensional domain, or implicity, as a map from 3 dimensions into a 1 dimensional variety. Certain surfaces require a parametric definition and other surfaces require an implicit definition. The parametric method may be used to define oriented or non-oriented surfaces of a single component. The implicit method can be used to describe surfaces of multiple components, which are orientable. Some surfaces like the sphere admit both representations.

When the subspaces of interest are manifolds of dimension m, the Whitney embedding theorem tells us that the euclidean covering manifold need not be of dimension larger than 2m+1.

The usual implicit surface is defined typically as the zero set of some function. Given the function, construct its gradient as a representation of the fiber normal to the surface. The idea is that there exist algebraic constructions of m-1 vectors that are transverse to the fiber. These m-1 vectors are defined as the associated or tangent vectors to the surface. The concept of m-1 associated vectors transversal (orthogonal) to a fiber can be developed even though the fiber is not defined as a gradient of a function.

Given an arbitrary 1-form on a space of dimension m it is always possible to construct algebraically m-1 vectors transversal to the 1-form by means of the interior product.

 Connection 1-forms, Curvature-torsion 2-forms

 Cartan's Repere Mobile

 Implicit Surfaces

 Parametric Surfaces

 As I have time I will replace this under construction symbol with a link to a more detailed pdf or tex file download.