Differential geometry notes 1

by Emin Orhan

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 \mathbf{x}= [x_1(t), x_2(t),x_3(t)] for I: a \leq t \leq b. This defines a point set M in \mathbb{R}^3. t is called the parameter.  We make the following assumptions about this function:

  • x_i(t) for i=1,2,3 are r times continuously differentiable in I.
  • For every t \in I, there is an i such that \frac{dx_i}{dt} is different from zero.

A representation of \mathbf{x} satisfying these assumptions is called an allowable parametric representation. We may obtain different parametrizations using transformations such as:

t = t(t^{\ast})

The point set M is invariant under such transformations. We make the following assumptions about these transformations:

  • t is defined in the interval I^{\ast}: a^{\ast} \leq t^{\ast} \leq b^{\ast} and the range of values corresponding to the interval I^{\ast} lies in I.
  • t is r times continuously differentiable in I^{\ast}.
  • The derivative \frac{dt}{dt^{\ast}} is different from zero everywhere in I^{\ast}.

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 \mathbb{R}^3 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 M corresponds to multiple values of t, 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 I 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 t are restricted to any closed and bounded subinterval of I.

A curve is said to be closed if it possesses at least one allowable representation whose vector function is periodic: \mathbf{x}(t + \omega) = \mathbf{x}(t) for some \omega > 0.