Can closed curves be assumed unit speed?

Question

Andrew Pressley's Elementary Differential Geometry textbook claims on page $21$ that "...we can always assume that a closed curve is unit-speed and that its period is equal to its length."

In the previous paragraph, Pressley derives this assuming the curve $\gamma$ is regular. This is obvious since regularity is equivalent to the existence of a unit speed parametrization. However, the summary statement itself here excludes the "regularity" assumption.

Later on in Chapter $2$, Pressley wants to show the total signed curvature of a closed plane curve is an integer multiple of $2\pi$. However, he starts out the proof by seemingly adhering to the principle quoted above and assumes the closed plane curve is unit speed. No regularity assumption was made.

Am I going crazy here, or am I just being pedantic and is regularity just being implicitly assumed both in this proof and the quoted statement above?

Answer 1

Clearly a curve has a unit-speed reparametrization if and only if it is a regular curve.

On p. 21 the author proves the following two facts:

If $\gamma$ is a regular closed curve, a unit-speed reparametrization of $\gamma$ is always closed.

This shows that [the unit-speed reparametrization] $\tilde \gamma$ is a closed curve with period $ℓ(γ)$. Note that, since $\tilde \gamma$ is unit-speed, this is also the length of $\tilde \gamma$.

He then writes

In short, we can always assume that a closed curve is unit-speed and that its period is equal to its length.

It is obvious that this is just a sloppy summary of the preceding results. You are right, one should correctly state that

In short, we can always assume that a regular closed curve is unit-speed and that its period is equal to its length.

Corollary 2.2.5 says (again sloppily) that the total signed curvature of a closed plane curve is an integer multiple of $2π$.

In the proof of the Corollary only unit-speed curves are considered, and probably this reminded you of the "In short" statement.

However, Chapter 2 only deals with regular curves. Definition 2.1.1 introduces curvature only for unit-speed curves. This is generalized by

So far we have only considered unit-speed curves. If $γ$ is any regular curve, then by Proposition 1.3.6, $γ$ has a unit-speed parametrization $\tilde γ$, say, and we can define the curvature of $γ$ to be that of $\tilde γ$.

For plane curves the concept of signed curvature is introduced. This is first done for unit-speed curves and is then generalized to regular curves. The author never considers non-regular curves in Chapter 2, and this should dispel all doubts.

Answer 2

No, this is false if the curve is not regular. For example, the cuspidal cubic, which can be smoothly parametrized as $\gamma(t) = (t^2, t^3)$, cannot be smoothly parametrized with unit speed. You should assume regularity for this result.