# Properly Defining a Smooth Curve

I have seen many different definitions of what it means for a curve to be "smooth". In this question, for instance, a curve $\gamma \colon [a,b] \longrightarrow \mathbb{R^n}$ is defined to be smooth if all derivatives exist and are continuous. This seems reasonable and in-line with what it means for any manifold to be smooth.

See, however, Edwards Calculus of Several Variables where a smooth curve is defined (actually, he uses the word "path"):A curve $\gamma \colon [a,b] \longrightarrow \mathbb{R^n}$ is said to be smooth if the derivative $\gamma^{\prime}(t)$ exists, is continuous and if $\gamma^{\prime}(t) \neq 0 \;\; \forall t \in [a,b]$.

Ignoring the fact that Edwards is only concerned with $C^{1}$ curves as opposed to $C^{\infty}$ curves, that this definition requires the derivative to be nonzero obviously makes it different from the first definition.

Now, consider yet another definition, this time from Wade's Introduction to Analysis (3rd Edition): A subset $C$ of $\mathbb{R}^m$ is called a $C^p$ curve in $\mathbb{R}^m$ if and only if there is a nondegenerate interval $I$ and a $C^p$ function $\gamma \colon I \longrightarrow \mathbb{R}^m$ that is injective on the interior of $I$ and $C = \gamma(I)$

Ignoring the fact that Wade is defining a curve as a set of points instead of its parametrization, this definition also differs from the first two in that $\gamma$ is required to be injective.

So, finally, here is my question: Is there a definition, or set of definitions, that could bring some consistency to this terminology? Perhaps there's a special name for a "curve" that is "injective" in addition to being "smooth"? Or, maybe there's a special name for a curve that never evaluates to the $0$-vector anywhere on it's domain? It seems to me that these are different attributes and should probably have distinctive nomenclature.

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My guess is that this just has to do with what theorems the author wants to present. It is somewhat related to the difference between types of submanifolds: immersed, embedded, or submersed. I don't think any of your definitions match perfectly with these, but you might want to glance through some properties of these to get an idea of the differences the hypotheses give. – Matt Dec 19 '11 at 15:32
not gonna happen. the idea of a map $I\to M$ is too simple and ubiquitous with varied uses for uniform terminology; you'd need like 50 adjectives for each instance. You could say things like "nonsingular $C^2$ which can be extended to a slightly larger open interval" etc, but after saying this once, you don't want to say it again. – yoyo Dec 19 '11 at 15:33
It depends on what is in the focus. If it is geometry, 'smooth' often means $C^\infty$. If it is differential equations, analysis of regularity questions becomes important, and one tries to reduce the regularity assumptions as far as possible. This allows to apply the resulting theorems in a wider context. The question of (non)zero derivatives for parametrizations is relevant if you want local injectivity, so objects will actually look smooth locally. When you aim at embeddings, you try to avoid defining objects using maps, since proving embeddedness may then become a pain or even impossible. – user20266 Dec 19 '11 at 16:13

Yes, there is a set of definitions that can bring consistency to the terminology.

A curve $\gamma\colon I \to \mathbb{R}^n$ is smooth iff it is $C^\infty$ (or $C^p$ for some authors).

A curve $\gamma\colon I \to \mathbb{R}^n$ is an immersion iff it is $C^\infty$ and $\gamma'(t) \neq 0$.

A curve $\gamma\colon I \to \mathbb{R}^n$ is simple iff $\gamma$ is injective on the interior of $I$.

(The reason for requiring injectivity on the interior, rather than on the whole interval, is so that we may legitimately speak of "simple closed curves." That is a simple closed curve $\gamma\colon [a,b] \to \mathbb{R}^n$ is a map that is injective on $[a,b)$ and has $\gamma(a) = \gamma(b)$.)

Granted, these definitions aren't 100% universal, but I think they're fairly standard. The term "regular" is sometimes used as a synonym for "immersion," but then again, "regular" can mean other things in other contexts, too (which is why I prefer to avoid the term altogether if I can help it).

And as you point out, some authors refer to a curve as the image of such a map, rather than the map itself.

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Your definitions of smooth and simple make sense; I'm not sure though how your definition of "immersion" accords with other definitions of that term. For example, Wikipedia says that an immersion is a map whose derivative is injective at every point in its domain. – ItsNotObvious Dec 19 '11 at 17:00
@3Sphere When the domain is $1$-dimensional, being an injective linear map is the same as being non-zero. – Dylan Moreland Dec 19 '11 at 17:05