What is the meaning of "infinitesimal structure"?

Question

Reading a Differential Geometry book I found this sentence: "A main theme in analysis on metric spaces is understanding the infinitesimal structure of a metric space." I cannot understand the meaning of infinitesimal structure, somebody can help me?

Answer 1

It means structure at an infinitesimal scale, yes?

OK, that probably wasn't terribly informative. It might help first to understand what might be meant by structure here. Intuitively, we might understand it as the way that distances and angles are related to each other across a space.

For instance, if you were to characterize a hill in a meadow, you might say that it's flat all around the hill, is "concave upward" at its foot, and is "concave downward" at its peak. In between is some other kind of curvature. In fact, we could say that the entire hill is described by its curvature.

Interesting fact: The hill is embedded in the three dimensions of ordinary space, but we can describe its curvature without any reference to this embedding, at all. We don't need to speak of "concave upward" or "concave downward" or any of that sort of directional talk; instead, we talk about how distances and angles change as we move smoothly over the hill.

At first, this idea seems a bit strange. In the first place, how do distances and angles change? And what are distances and angles over a curved surface anyway?

We might approach these questions by imagining an array of ants approaching the hill from a distance. These ants are programmed to walk in as straight a line as possible (in a sense that can be made precise). Initially, we might conceive of them as being arrayed in a perfect grid, but as they traverse the curved hill, it is impossible for them to maintain that perfect grid. The distances between the ants—the shortest possible string that can be run between a pair of ants along the surface of the hill—and the angles between those strings are constantly changing as they go over the hill.

Note that the way these things change are entirely independent of how the hill is oriented. Someone monstrous could take the whole hill and its surroundings out of the earth, and hold it upside-down, and those distances and angles would change the same way (assuming the ants to be able to continue walking as they had before). They are intrinsic to the curvature of the hill—its structure, in other words. And if you imagine the ants getting infinitesimally small and infinitesimally close together, you obtain the infinitesimal structure of that hill.

That is one of the things differential geometry is about, at its most basic level.

ETA: Another comment on the difference between curvature and embedding. If the hill were a long one-dimensional "roll" (like a speed bump), then even though that hill might be embedded in a curved way, it actually has no intrinsic curvature. We can verify this by imagining the ants again. They start in a perfect grid, and they can go over the hill in a perfect grid, so long as we measure distances along the surface of the hill, using those strings again. The angles between the strings do not change, either. Hence, from the perspective of an ant, the structure of this hill is the same as that of plain ol' flat ground.