Differential geometry notes 1
These are some notes on differential geometry I took while studying Erwin Kreyszig’s Differential Geometry. I will keep posting these notes as I read the book.
Consider the vector function for . This defines a point set in . is called the parameter. We make the following assumptions about this function:
- for are times continuously differentiable in .
- For every , there is an such that is different from zero.
A representation of satisfying these assumptions is called an allowable parametric representation. We may obtain different parametrizations using transformations such as:
The point set is invariant under such transformations. We make the following assumptions about these transformations:
- is defined in the interval and the range of values corresponding to the interval lies in .
- is times continuously differentiable in .
- The derivative is different from zero everywhere in .
A transformation satisfying these assumptions is called an allowable parametric transformation. Allowable parametric representations can be grouped into equivalence classes by means of allowable parametric transformations: two allowable representations are equivalent if there is an allowable transformation transforming one into the other.
Definition 6.1: A point set in which can be represented by the allowable representations of an equivalence class is called an arc of a curve.
Multiple point: If a certain point of corresponds to multiple values of , it is called a multiple point of the arc. An arc having no multiple points is called simple.
Definition 6.2: A point set is called a curve if it can be represented by an equivalence class of allowable representations whose interval is not assumed to be closed or bounded, but which are such that one always obtains an arc of a curve if the values of the parameter are restricted to any closed and bounded subinterval of .
A curve is said to be closed if it possesses at least one allowable representation whose vector function is periodic: for some .