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In most undergraduate differential geometry courses -- I am thinking of do Carmo's "Differential Geometry of Curves and Surfaces" -- the topic of study is curves and surfaces in $\mathbb{R}^3$. However, the definition of "curve" and "surface" are usually presented in very different ways.

A curve is defined simply as a differentiable map $\gamma\colon I \to \mathbb{R}^3$, where $I \subset \mathbb{R}$ is an interval. Of course, some authors prefer to define a curve as the image of such a map, and others require piecewise-differentiability, but the general concept is the same.

On the other hand, surfaces are essentially defined as 2-manifolds.

Similarly, in graduate courses on manifolds -- I am thinking of John Lee's "Introduction to Smooth Manifolds" -- one talks about curves $\gamma\colon I \to M$ in a manifold, and can do line integrals over such curves, but talks separately about embedded/immersed 1-dimensional submanifolds.

My question, then, is:

Why make (parametrized) curves the object of study rather than 1-manifolds?

Earlier, I asked a question that was perhaps meant to hint at this one, though I didn't say so explicitly.

Ultimately, I would simply like to say "curves are 1-manifolds and surfaces are 2-manifolds," and am looking for reasons why this is correct/incorrect or at least a good/bad idea. (So, yes, I'm looking for a standard definition of "curve.")

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I maybe misinterpreting your question, but there is some pedagogical value in treating curves before surfaces, and embedded objects in $\mathbb{R}^3$ (or evene in $\mathbb{R}^2$) before abstract manifolds. – timur Dec 24 '10 at 5:52
up vote 18 down vote accepted

Essentially because the connected 1-manifolds are ("up to...") $(0,1)$ and $S^1$, so the notion of curve captures all of the possibilities of 1-manifolds sitting in higher-dimensional manifolds. In other words, the situation for 1-dimensional manifolds is so simple that it really makes no sense to use the full machinery of embeddings and immersions to talk about them other than just checking that the definitions of embedding and immersion are compatible with the definition of curve.

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Also, in agreement to what kahen said:

In a regular parametrization, you can always find a metric that is constant along the curve leading to zero intrinsic curvature. This is also a reason why it might be somehow not very enlightening to discuss curves as one dimensional manifolds in its own right.

Curves, on the other hand, can have extrinsic curvature which can for some body moving along a trajectory be interpreted as the acting force.



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