Curves & Parameters on a Manifold M

[Rasalhague]

A smooth curve, [itex]C[/itex], on a manifold [itex]M[/itex] is simply a [itex]C^\infty[/itex] map of [itex]\mathbb{R}[/itex] (or an interval of [itex]\mathbb{R}[/itex]) into [itex]M, \enspace C:\mathbb{R}\to M[/itex] (Wald: General Relativity, p. 17).

A curve on a manifold [itex]M[/itex] is a smooth (i.e. [itex]C^\infty[/itex]) map from some interval [itex](-\epsilon,\epsilon)[/itex] of the real line into [itex]M[/itex] (Isham: Modern Differential Geomtry).

These definitions seem to suggest that the same subset of [itex]M[/itex] could be the range/image/arc of many different curves, each having a different parameter. Is that right, or should I think of a curve as an equivalence class of such maps?

 

[Hurkyl]

Yes and no.

For a great many purposes, reparametrizing a curve doesn't change anything, and so you can consider an equivalence class under reparametrization.

Some applications might not be that permissive and you would need to take care. Others (e.g. homotopy, or line integrals of complex analytic functions) might be even more permissive allowing wider equivalence classes.


As for the point set, one rather important feature of a curve that might not be detectable from its trace is that a curve may retrace the same points. I think for most applications, a curve that goes once around the circle and a curve that goes twice around the circle will be different.

Orientation is another matter that isn't detectable from the point set.

 

[Rasalhague]

Thanks, Hurkyl. Isham says the word curve should be reserved for the map itself, as opposed to its image/range/trace. Is there a standard (or preferable) term for the sort of point set (one-dimensional submanifold?) that's colloquially called a curve, a word which doesn't specify a parameterisation? Path perhaps, or oriented path (if an orientation is given)? In the context of general relativity, timelike curves in the sorts of spacetime we have experience of don't retrace the same points, so I guess that wouldn't be an issue there. And references to "closed timelike curves" seem to treat these as (hypothetical) non-orientable paths, in which case I suppose the word curve, in Isham and Wald's sense, wouldn't really apply, since any given map gives an orientation, doesn't it? In fact, would "non-orientable curve" be a contradiction in terms and "oriented curve" a tautology?

In the context of general relativity, being a geodesic is usually said to be a property of a curve, although perhaps curve is to be taken there in its colloquial sense or as the directed trace of an equivalence class of curves having the same unit tangent vector field. Otherwise, could the same future-directed timelike path be paramaterised with one parameter as a geodesic and, with another parameter, as a non-geodesic curve? I'm guessing not, as that seems to go against the spirit of defining physical properties independently of coordinate representation.