Differential Geometry/Basic Concepts

Consider n functions, x₁(t), x₂(t), x₃(t), ..., x_n(t). Then consider the vector f function in Rⁿ which is given by f(t)=(x₁(t), x₂(t), x₃(t), ..., x_n(t)) is called a parametric representation of a set M, where M is the image of the function.

The point sets which shall be considered here must be able to be represented by a parametric representation (called an allowable parametric representation) f(t) where:

1. The function f is continuously differentiable.
2. The derivative f'(t) is different from the null vector for all t.

What is most important here are not individual parametric representations of vector function, but rather equivalence classes of vector functions which are have the same properties. These equivalences classes are called arcs.

Two of parametric representations f₁(x) and f₂(x) of a set are said to be equivalent if

1. f₁⁻¹(f₂(t)) is defined on a closed interval [a,b].
2. f₁⁻¹(f₂(t)) is continuously differentiable.
3. f₁⁻¹(f₂(t))≠0 for all t in [a,b]

A point in a vector function f(t₁)=f(t₂) (t₁≠t₂) is called a multiple point. Arcs without multiple points are simple.

Arcs themselves are continuous as a vector function. Let $\epsilon >0$. Since each $x_{k}$ is continuous, there exists for each $k$ a $\delta _{k}$ such that

$\left|x_{k}(t_{1})-x_{k}(t_{2})\right|<{\frac {\epsilon }{\sqrt {n}}}\quad {\text{if}}\quad \left|t_{1}-t_{2}\right|<\delta _{k}$

Let δ = min{δ_k}. Then when |t₁-t₂|<ε, |f(t₁)-f(t₂)|=${\sqrt {\textstyle \sum _{k=1}^{n}(x_{k}(t_{1})-x_{k}(t_{2}))^{2}}}$<ε.

All simple arcs are thus homeomorphic to a segment.

Finally, curves are point sets representable by a arc such that for all functions within the arc, there is an interval whose image is the curve and such that any subinterval is an allowable parametric representation of some set.

A closed curve is one that is representable by a periodic function. All closed curves are homeomorphic to the circle.