Isn't a sphere in hyperbolic 3-space isometric to a sphere in Euclidean 3-space? So the sphere has *positive* constant Gaussian curvature, and the circles have circumference 2 pi k sin(r/k)? [Certainly, if the sphere has constant Gaussian curvature -- as you deduced from your symmetry argument -- then it must be positive by global Gauss- Bonnet.] -- APG.
Sent: Monday, February 19, 2018 at 7:30 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Curved space paradox
The following intuitive line of reasoning leads to a contradictory conclusion, and I don't know what's wrong with it. Obviously one of my assumptions or beliefs about spaces with constant Gaussian curvature is wrong, but I can't figure out which.
Suppose we have a 3-dimensional space of constant, negative Gaussian curvature.
For some large enough R, construct the locus of points at distance R from a fixed center C. This forms a 2-dimensional surface which is topologically an ordinary sphere. Select a point P on this sphere, and begin drawing circles of increasing radius r on the sphere centered at P. I mean r to be measured along the surface of the sphere, not straight through the hyperbolic 3-space in which the sphere is embedded.
Now, this sphere must itself have constant Gaussian curvature; I don't have a fast proof but surely there is a symmetry argument that can establish it.
Watch how the circumference of a circle, centered at P, changes with its radius r. For small r, it looks like 2 pi r. But as r gets larger, the negative curvature of the ambient hyperbolic space begins to make itself felt, and the circumference goes above 2 pi r, by some factor related to sinh. (If it doesn't, you didn't make R big enough -- that is exactly what I meant by "large enough R" above.)
But later, as r becomes commensurate with R, the circumference rises to a maximum and begins to shrink again, because the sphere itself has finite circumference.
On a surface of constant curvature, the circumference should either fall below 2 pi r, or it should exceed it forever. It should not do both.
Surely I am not the first person to think these confusing thoughts. What is wrong with my reasoning? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun