Why research "curves" as opposed to 1 - manifolds?

In the majority of undergraduate differential geometry training courses - - I am considering do Carmo is "Differential Geometry of Curves and also Surfaces" - - the subject of research is contours and also surface areas in $\mathbb{R}^3$. Nonetheless, the definition of "curve" and also "surface" are generally offered in really various means.

A contour is specified merely as a differentiable map $\gamma\colon I \to \mathbb{R}^3$, where $I \subset \mathbb{R}$ is a period. Certainly, some writers favor to specify a contour as the photo of such a map, and also others call for piecewise - differentiability, yet the basic principle coincides.

On the various other hand, surface areas are basically specified as 2 - manifolds.

In a similar way, in graduate training courses on manifolds - - I am thinking about John Lee is "Introduction to Smooth Manifolds" - - one speaks about contours $\gamma\colon I \to M$ in a manifold, and also can do line integrals over such contours, yet talks independently concerning embedded/immersed 1 - dimensional submanifolds.

My inquiry, after that, is :

Why make (parametrized) contours the object of research as opposed to 1 - manifolds?

Previously, I asked a question that was probably suggested to mean this set, though I really did not claim so clearly.

Inevitably, I would merely such as to claim "curves are 1 - manifolds and also surface areas are 2 - manifolds, " and also am seeking reasons that this is correct/incorrect or at the very least a good/bad suggestion. (So, yes, I'm seeking a typical definition of "curve.")

2019-12-02 02:47:27
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Answers: 2

Essentially due to the fact that the attached 1 - manifolds are ("up to ...") $(0,1)$ and also $S^1$, so the idea of contour records every one of the opportunities of 1 - manifolds being in greater - dimensional manifolds. To put it simply, the scenario for 1 - dimensional manifolds is so straightforward that it actually makes no feeling to make use of the complete equipment of embeddings and also immersions to speak about them apart from simply examining that the interpretations of embedding and also immersion work with the definition of contour.

2019-12-03 04:11:39

Also, in arrangement to what kahen claimed :

In a normal parametrization, you can constantly locate a statistics that is constant along the contour bring about absolutely no inherent curvature. This is additionally a reason that it could be in some way not really informing to review contours as one dimensional manifolds in its very own right.

Contours, on the various other hand, can have external curvature which can for some body relocating along a trajectory be taken the acting pressure.



2019-12-03 03:29:28