### 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 $\mathbf{x}= [x_1(t), x_2(t),x_3(t)]$ 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 $\omega > 0$.