On Monday, February 19, 2018, 1:01:00 PM PST, Andy Latto <andy.latto@pobox.com> wrote: On Mon, Feb 19, 2018 at 2:30 PM, Allan Wechsler <acwacw@gmail.com> wrote:
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.)
I'm not convinced of this. Why do you think the positive curvature of the sphere won't have a greater effect than the negative curvature of the ambient space? Both effects depend on R, so I don't see why "make R large" guarantees that one is a greater effect than the other. I may be missing something, but I think at least you need to fill i some detail here to sharpen the paradox. Andy The hyperbolic 3-space can be given the metric ds^2 = dr^2 + sinh^2 r (dθ^2 + sin^2 θ dφ^2). The 2-sphere of constant r is a plain ordinary 2-sphere of radius sinh r, and we may ignore that it happens to be embedded in a hyperbolic space. -- Gene