What is the meaning of "without referring back to the ambient space $R^3$ where the surface lies"?

Question

I'm reading Do Carmo's Differential Geometry book, here:

What is the meaning of "without referring back to the ambient space $R^3$ where the surface lies"? Does it mean that we can compute it directly without appealing to some inverse mapping? If so, why don't we need to refer back?

This concerns me because it seems that there are situations where we need to refer back to the ambient space and situations where we don't and up to now, It's not clear where we need it and where we don't, I just accepted the definitions and constructions without deeper thought because I still can't.

This gives me a lot of curiosity, for example: In a definition I mentioned in a previous question:

I noticed that this is defined referring back to a map between open sets in $R^2.$ Can we make this definition without this?

Answer 1

tl; dr: It's arguably impossible to answer a philosophical question about definitions, but if this question came up during a chat over beverages, I'd say

1. We know how to do calculus on (non-empty open subsets of) Cartesian spaces.
2. The definition of a smooth manifold uses this knowledge to extend the concept of smoothness.
3. We know how to use calculus to measure arc length, area, and so forth, in a manifold embedded in a Euclidean space.
4. A certain amount of ambient Euclidean structure can be restricted to an embedded manifold, and then viewed as data intrinsic to the manifold, separately from a particular embedding.

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Regarding point 2., we use coordinate charts (or parametrizations, their mapping inverses) to transfer questions about functions and mappings on manifolds to questions about functions and mappings on Cartesian space. In order for the concepts such as smoothness to be well-defined (independent of chart) we impose a compatibility condition between parametrizations $\mathbf{x}_{1}$ and $\mathbf{x}_{2}$ amounting to smoothness of $\mathbf{x}_{2}^{-1} \circ \mathbf{x}_{1}$.

For point 4., thinking specifically of surfaces in Euclidean three-space, a parametrization induces three functions $E$, $F$, and $G$ defined in a coordinate neighborhood (open subset of the plane). These suffice to specify the lengths of tangent vectors, the angle between two non-zero tangent vectors at a point, the arc length of a piecewise-smooth path, and the area of a region of surface. These functions and their derivatives also suffice to define geodesics, paths in the surface of locally shortest length. Remarkably (Gauss's Theorema Egregium) these functions and their derivatives also detect whether or not a surface is locally isometric to the Euclidean plane in the sense that small geodesic triangles have total interior angle $\pi$. Our ability to define and measure these quantities using functions of two variables defined in a coordinate neighborhood is what do Carmo means by "without referring back to the ambient space." Analogously, general relativity allows us to conceptualize and work with the geometry of spacetime without imagining our universe embedded in a higher-dimensional space.

By contrast, a parametrization, particularly an embedding of a surface in Euclidean three-space, also induces components $e$, $f$, $g$ (or $\ell$, $m$, $n$ depending on the author) of a second fundamental form. Loosely, these functions measure how the surface bends in the ambient space, or more precisely, how a continuous unit normal field varies in coordinates. Because a unit normal field "does not lie in the surface," geometers think of the second fundamental form as "extrinsic" geometry that does refer to the ambient space.

Answer 2

They key difference between your two examples is the dimension of the space. The manifold is a 2-dimensional surface in both cases. In the first case this surface is embedded into the 3-dimensional space $\mathbb{R}^3$.

In the second case you consider the parametrizations which map [parts of] the 2-dimensional space $\mathbb{R}^2$ to [parts of] the surface. On these parts the parametrization maps or charts are bijections.