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In his book, Differential Geometry of Curves and Surfaces, Do Carmo defines curves to be maps, while he defines (regular) surfaces to be sets rather than maps.

Is there a particular reason he chose to define them this way? That is, why are curves not defined as sets, and vice versa? Must surfaces be defined as sets because you cannot always construct a surface with one particular 'map' but rather, patch together multiple maps to cover the said surface in a smooth manner (and with no intersections)?

Thanks

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You can also define curves as sets, but in the end you cannot escape using functions to describe a curve.

Here is a definition, mimicking the definition of surfaces:

A curve $C\subseteq\Bbb R^3$ is a subset of $\Bbb R^3$ satisfying the following property:

For every point $p\in C$ there exists an open neighbourhood $U\subseteq\Bbb R^3$ of $p$, an open set $V\subseteq\Bbb R$, and a smooth function $f:V\to U$ such that

  1. $f:V\to U\cap C$ is a homeomorphism.

  2. For each $t\in V$, the derivative $f'(t)$ is not the zero vector.

As you can see, the function $f$ would essentially be the function that defines the curve.

You can define parametric surfaces like how you define parametric curves. You can say that a parametric surface is a smooth function $f:V\to\Bbb R^3$ from an open set $V\subseteq\Bbb R^2$. If the Jacobian matrix of $f$ is injective everywhere in the domain $V$, $f$ is said to be regular. This definition appears on page 80 of do Carmo's Differential Geometry of Curves and Surfaces, second edition.

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  • $\begingroup$ Found it! Thanks $\endgroup$ – T J. Kim Apr 16 '18 at 18:05

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