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Hello I'm not sure if I used the correct terms but imagine that we have this sheet of paper with these two points: image they are the opposite side of the paper.

But if now we rolled it up into a cylinder (taking the two red points as extremities) they would be really close therefore their distance isn't the same as it was when they were on the sheet of paper, or I'm probably misunderstanding something so could someone clear it up for me please?

Gr_Fan
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  • "intrinsic" means you aren't allowed to leave the piece of paper. All you know are the geometric properties (lengths of curves, geodesics, angles, etc.) that lie inside the piece of paper.

    Even then, a cylinder is the not the same as a sheet of paper, because you can travel along a geodesic in the cylinder and return to the same point. So maybe it's best to think about a half cylinder.

    If you consider curves that leave the paper and measure distances via lines that lie at least partially outside the paper, then that's called extrinsic geometry.

     – Deane Oct 16 '17 at 16:52
  • Only for points away from the edge you are joining. Perhaps it is better to say that the intrinsic properties of a straight sheet of paper are the same as the properties of a gently bent sheet of paper. – orangeskid Oct 16 '17 at 17:21
  • Thanks everyone especially orangeskid and deane I think I got it now – Gr_Fan Oct 17 '17 at 11:07

3 Answers3

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Intrinsic properties of the plane and the cylinder are NOT the same.

LOCAL intrinsic properties are the same. This means exactly the following: for every point on the cylinder there is a (small) neighborhood about this point which is isometric to some subset of the plane. Isometric means that there is a one-to-one map preserving distances. Distances are measured with a rope which must be completely contained in the cylinder when we measure on the cylinder, and on the plane when we measure in the plane.

Alexandre Eremenko
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Some of the geometric features one likes to equip surfaces with are notions like:

  • Which curves are straight?
  • What is the area of some region?

For a counter-example, consider this map of the spherical surface of the Earth (taken from a google search):

Great circle drawn on a map of the Earth

Despite appearances, the pink line is a straight line, and Greenland does not have an area rivaling that of the continental United States! The usual geometry on a sheet of paper does not share these geometric features with the surface of the Earth!

The story is different with the usual cylinder. A line drawn on a cylinder is straight if and only if it is straight when you unroll it to become a sheet of paper. The area of a region drawn on a cylinder? The same as the area when unrolled. Lengths of curves and angles between lines? Still the same after unrolling.

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To go in a slightly different direction than the previous answers, it is perhaps instructive to think of the cylinder as a quotient of $\mathbb{R}^{2}$ and to realize the geometry of the cylinder as descending from the quotient map.

Specifically, taking standard Cartesian $xy$-coordinates on $\mathbb{R}^2$, define an equivalence relation on points $P(x_0, y_0)$ by $P(x_0, y_0) \sim Q(x_{1}, y_{1})$ if and only if

  • $y_{0} = y_{1}$, and
  • $x_{1} - x_{0}$ is an integer multiple of $2\pi R$, where $R$ is the desired radius of your cylinder (i.e., $x_{1} - x_{0} = n2\pi R$, for some $n \in \mathbb{Z}$).

Now thinking of the geometry of the cylinder as being the geometry induced by the quotient map, the local nature of the intrinsic geometries being the same (and the failure of the global intrinsic geometries to be the same) is hopefully clear.

This construction is helpful as it is the traditional way in which one realizes a flat metric on the torus, and when one replaces $\mathbb{R}^{2}$ with the hyperbolic plane $\mathbb{H}^2$, it generalizes to the standard construction of how one puts a metric of constant negative curvature on a compact surface of genus $g \ge 2$.

THW
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