What is a curve? (Definition)

Question

Studing for an introductory differential geometry course I've come across a lot of variations for the definition of a curve in $\mathbb{R}^3$, ranging from the most general one, admiting any continuous function on a real interval $I$ into $\mathbb{R}^3$ as a curve, to very narrow ones defining it as a finite union of the graphs of regular curve parametrizations (or something like that). Do we have a real teminlogy issue, or there is some universally accepted standard?

Answer 1

$\newcommand{\Reals}{\mathbf{R}}$To flesh out the comments: There's not a standard convention/definition for a curve, because curve-like sets and mappings play such a variety of roles in geometry and analysis. The definitions below are not universal and definitive, but are intended to serve as a "cultural roadmap".

Definition: Let $(X, T)$ be a topological space. A (real) path in $X$ is a continuous mapping $\gamma$ from an interval $I$ of real numbers (with the Euclidean topology) into $(X, T)$.

Qualitative uses of paths include:

• Topology: Deforming one mapping to another (as in algebraic topology, where a path in a space of mappings is a homotopy, and under which quantities such as residues and winding number are invariant).

• Smooth Structure: Probing a space (as in the definition of a tangent vector to a smooth manifold as an equivalence class of paths; taking normal sections of a surface embedded in Euclidean three-space).

• Differential Geometry: Measuring a space (as when defining a topological metric on a Riemannian manifold by taking the infimum of lengths of paths; or carrying an orthonormal frame by parallel transport).

• Kinematics: Modeling the location of a particle or system state as a function of time (as when solving an ordinary differential equation).

These distinctions are overlapping and somewhat artificial.

---

Technical conditions on paths

• If $I$ is compact, the image $\gamma(I)$ is compact, which is often topologically desirable. On the down side, an interval is not a manifold but a manifold with boundary, and "differentiability" at an endpoint can be vexing (Note 1).

  When differentiability matters, one may assume $I$ is an open interval. Unfortunately, open intervals are not compact, and even a real-analytic path can be...pathological (Note 2).

  When one needs both compactness and differentiability, it's customary to assume $\gamma$ is defined on some compact interval $I$, but that $\gamma$ has a smooth extension to some larger open interval.

• If $(X, T)$ is a smooth manifold (i.e., is equipped with a coordinate atlas whose overlap maps are $k$-times continuously differentiable for some $k \geq 1$, or infinitely-differentiable, or real-analytic), one often works with smooth paths.

  Particularly in differential geometry, one usually assumes a path is regular, i.e., has non-vanishing derivative, so the image has a tangent line at each point.

  Very often when differential topologists and geometers speak of a curve, they mean an image of a regular path. (Thus, a path is a mapping, while a curve is a set of points.)

  For technical reasons involving concatenation of curve segments, it's often simpler to work with piecewise-smooth paths, for which the domain $I$ can be divided into finitely many subintervals, on each of which $\gamma$ is regular and the derivative $\gamma'$ has one-sided limits at the endpoints.

• A path may be simple (injective, i.e., not self-crossing) or closed (having compact domain $I = [a, b]$ and satisfying $\gamma(a) = \gamma(b)$. (In smooth settings, one may also assume derivative conditions such as $\gamma'(a) = \gamma'(b)$.)

• A path $\gamma:[a, b] \to (X, d)$ in a metric space is rectifiable (of finite length) if, letting $(t_{i})_{i=0}^{N}$ denote an arbitrary partition $a = t_{0} < t_{1} < \cdots < t_{N} = b$ and taking the supremum over all partitions, $$ \sup \sum_{i=0}^{N} d(\gamma(t_{i-1}), \gamma(t_{i})) < \infty. $$ A piecewise-$C^{1}$ path (on a compact interval) in a Riemannian manifold is rectifiable, but a continuous bijection need not be locally rectifiable at any point. [Note 5.]

---

Notes and counterexamples

1. The logarithmic spiral $$ \gamma(t) = \begin{cases} e^{-1/t}(\cos(1/t), \sin(1/t)) & 0 < t \leq 1, \\ (0, 0) & t = 0, \end{cases} $$ is (uniformly!) continuous, and real-analytic in $(0, 1)$, but winds infinitely many times around the origin in every neighborhood of $0$.

2. Fix an irrational number $\alpha$, and define the irrational winding of slope $\alpha$ to be the curve $$ \gamma(t) = (\cos t, \sin t, \cos(\alpha t), \sin(\alpha t),\quad -\infty < t < \infty. $$ The image of $\gamma$ lies in the (compact) torus $S^{1} \times S^{1} \subset \Reals^{4}$, and it's straightforward to check that $\gamma$ is regular and injective. The image of $\gamma$ is dense in the torus, however, and the complement of the image is fantastically complicated, being topologically connected, but having a continuum of path components. (The space of path components is closely related to the standard construction of a Lebesgue non-measurable set.)

3. A smooth (even real-analytic) path can have non-smooth image if the path is not regular. Standard examples include:

   • The cusp $\gamma(t) = (t^{3}, t^{2})$.

   • The astroid $\gamma(t) = (\cos^{3} t, \sin^{3} t)$.

   • Hypocycloids.

4. If $I = [0, 1]$, there exists a continuous surjection $\gamma:I \to I \times I$, namely, a path whose image is the unit square. Obvious iterative constructions give paths whose image is an arbitrary box $$ [a_{1}, b_{1}] \times \cdots \times [a_{n}, b_{n}] \subset \Reals^{n}. $$ No such path can be injective, since a continuous bijection whose domain is a compact Hausdorff space is a homeomorphism to its image.

5. The Koch snowflake is the image of a continuous bijection $\gamma:S^{1} \to \Reals^{2}$. If $p$ and $q$ are arbitrary points, the length of either portion of the snowflake "between" $p$ and $q$ is infinite.